Question

Find the sum of the series 1.2 + 2.3x + 3.4x^2 + ..... infinity

Answer 1

Ello❤

$a{}(n) = (nth \: term \: of \: 1.2.3.4...) \times (nth \: term \: of2.3.4... \\ a(n) = a + (n - 1)d \\ d = 1 \\ a(n) = (1 + (n - 1)1) \times (2 + (n - 1)1) \\ \: \: \: \: \: \: \: \: \: \: \: = (1 + n - 1) \\ a(n) = n \times (1 + n) \\ a(n) = n + {n}^{2} \\ \\ s(n) = ( \frac{n(n + 1)}{2} ) + \frac{n(n + 1)(2n + 1)}{6} ) \\ = \frac{n(n + 1}{2} ( \frac{3+ 2n + 1}{3} ) \\ = \frac{n(n + 1}{2} ( \frac{4 + 2n}{3} ) \\ = \frac{n(n + 1}{2} 2( \frac{2 + n}{3} ) \\ = \frac{n(n + 1)(2 + n)}{3}$

Hope its help u❤

Thanks ✴✴