Question

plz solve the questions​

Answer 1

Given,

$a_{n}=\frac{2^{n-1}}{5^{n-2}}\\=\frac{2^{n-1}}{5^{n-1}\times5^{-1}}}\\=5(\frac{2}{5})^{n-1}$

Which is of the form $ar^{n-1}$, where $a=5, r=\frac{2}{3}$

Indeed this sequence is a G.P

Now,

Using the G.P summation to kth terms formula,

$S_k=\frac{a(r^k-1)}{r-1}\\\\=5\frac{(\frac{2}{5})^k-1}{\frac{2}{5}-1}\\\\=5\frac{(\frac{2}{5})^k-1}{\frac{2-5}{5}}\\\\=\frac{25}{3}[1-(\frac{2}{5})^k]$

Then,

$S_\infty=\lim_{k\to\infty}\frac{25}{3}[1-(\frac{2}{5})^k]\\\\=\frac{25}{3}\lim_{k\to\infty}[1-(\frac{2}{5})^k]\\\\=\frac{25}{3}[1-\lim_{k\to\infty}(\frac{2}{5})^k]\\\\=\frac{25}{3}[1-0]\\\\\left[(\frac{2}{5})^k\to\:0\:as\:k\to\infty\right]\\\\=\frac{25}{3}\\\\\therefore S_\infty=\frac{25}{3}$