If 4 + \(\frac{4+d}{5}\) + \(\frac{4+2d}{5^2}\) + .......∞ = 10, the

1.	If 4 + \(\frac{4+d}{5}\) + \(\frac{4+2d}{5^2}\) + .......∞ = 10, then d is equal to (a) 5 (b) 8 (c) 10 (d) 16
Answer» (d) 16 Let S = 4 + \(\frac{4+d}{5}\) + \(\frac{4+2d}{5^2}\) + \(\frac{4+3d}{5^3}\)+ ........∞ ∴ \(\frac{1}{5}\)S = \(\frac{4}{5}\) + \(\frac{4+d}{5^2}\) + \(\frac{4+2d}{5^3}\)+ ........∞ ⇒ S - \(\frac{1}{5}\)S = 4 + \(\frac{d}{5}\)\(\bigg[1+\frac{1}{5}+\frac{1}{5^2+}.......\infty\bigg]\) ⇒ \(\frac{4}{5}\)S = 4 + \(\frac{d}{5}\)x \(\frac{1}{1-\frac{1}{5}}\) ⇒ \(\frac{4}{5}\)S = 4 + \(\frac{d}{5}\) x \(\frac{5}{4}\) ⇒ \(\frac{4}{5}\)S = 4 + \(\frac{d}{4}\) ⇒ S = 5 + \(\frac{5d}{16}\) Given S = 10 ∴ 5 + \(\frac{5d}{16}\) = 10 ⇒ \(\frac{5d}{16}\) = 5 ⇒ d = 16.

If 4 + \(\frac{4+d}{5}\) + \(\frac{4+2d}{5^2}\) + .......∞ = 10, then d is equal to (a) 5 (b) 8 (c) 10 (d) 16

Answer»

(d) 16

Let

S = 4 + \(\frac{4+d}{5}\) + \(\frac{4+2d}{5^2}\) + \(\frac{4+3d}{5^3}\)+ ........∞

∴ \(\frac{1}{5}\)S = \(\frac{4}{5}\) + \(\frac{4+d}{5^2}\) + \(\frac{4+2d}{5^3}\)+ ........∞

⇒ S - \(\frac{1}{5}\)S = 4 + \(\frac{d}{5}\)\(\bigg[1+\frac{1}{5}+\frac{1}{5^2+}.......\infty\bigg]\)

⇒ \(\frac{4}{5}\)S = 4 + \(\frac{d}{5}\)x \(\frac{1}{1-\frac{1}{5}}\) ⇒ \(\frac{4}{5}\)S = 4 + \(\frac{d}{5}\) x \(\frac{5}{4}\)

⇒ \(\frac{4}{5}\)S = 4 + \(\frac{d}{4}\) ⇒ S = 5 + \(\frac{5d}{16}\)

Given S = 10

∴ 5 + \(\frac{5d}{16}\) = 10 ⇒ \(\frac{5d}{16}\) = 5 ⇒ d = 16.