Buktikan dengan menggunakan induksi matematika bahwa \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{n-1} n^{2}=\frac{(-1)^{n-1} n(n+1)}{2} \) untuk setiap \( n \in \mathbb{N} \)
Dalam matematika, metode induksi matematika adalah teknik yang digunakan untuk membuktikan pernyataan yang berlaku untuk setiap bilangan bulat positif. Dalam artikel ini, kita akan menggunakan metode induksi matematika untuk membuktikan rumus yang diberikan, yaitu \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{n-1} n^{2}=\frac{(-1)^{n-1} n(n+1)}{2} \) untuk setiap \( n \in \mathbb{N} \). Langkah 1: Basis Induksi Pertama, kita akan membuktikan rumus ini untuk \( n = 1 \). Ketika \( n = 1 \), rumus menjadi \( 1^{2} = \frac{(-1)^{1-1} 1(1+1)}{2} \), yang dapat disederhanakan menjadi \( 1 = 1 \). Oleh karena itu, rumus ini benar untuk \( n = 1 \). Langkah 2: Langkah Induksi Selanjutnya, kita akan menggunakan asumsi bahwa rumus ini benar untuk \( n = k \) dan membuktikan bahwa rumus ini juga benar untuk \( n = k + 1 \). Dalam hal ini, kita akan mengasumsikan bahwa \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2}=\frac{(-1)^{k-1} k(k+1)}{2} \) adalah benar. Kita akan menggunakan asumsi ini untuk membuktikan rumus ini untuk \( n = k + 1 \). Pertama, kita tambahkan \( (k+1)^{2} \) ke kedua sisi rumus yang kita asumsikan benar. Kita mendapatkan: \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1)}{2} + (k+1)^{2} \) Kita dapat menyederhanakan persamaan ini menjadi: \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1)}{2} + \frac{2(k+1)^{2}}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)^{2}}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2}+3^{2}-4^{2}+\cdots \cdots+ \) \( (-1)^{k-1} k^{2} + (k+1)^{2} = \frac{(-1)^{k-1} k(k+1) + 2(k+1)(k+1)}{2} \) \( 1^{2}-2^{2