If 5th term of a G.P is 32 and common ratio 2, then the sum of 14 term

1.	If 5th term of a G.P is 32 and common ratio 2, then the sum of 14 terms isA) 32,766 B) 16,388 C) 64,432 D) 30,746
Answer» Correct option is (A) 32,766 Let first term of G.P. be a. Given that common ratio is r = 2. \(\therefore5^{th}\) term of G.P. is \(a_5=ar^4\) \((\because a_n=ar^{n-1})\) \(\Rightarrow ar^4=32\) \(\Rightarrow a.2^4=32\) \(\Rightarrow a=\frac{32}{2^4}=\frac{32}{16}=2\) Sum of first 14 terms is \(S_{14}=\frac{a(r^{14}-1)}{r-1}\) \(\left(\because S_{n}=\frac{a(r^{n}-1)}{r-1}\right)\) \(=\frac{2(2^{14}-1)}{2-1}\) \((\because a=2\;\&\;r=2)\) = 2 (16384 - 1) \((\because2^{14}=16384)\) = 32766 Correct option is A) 32, 766

If 5th term of a G.P is 32 and common ratio 2, then the sum of 14 terms isA) 32,766 B) 16,388 C) 64,432 D) 30,746

Answer»

Correct option is (A) 32,766

Let first term of G.P. be a.

Given that common ratio is r = 2.

\(\therefore5^{th}\) term of G.P. is \(a_5=ar^4\) \((\because a_n=ar^{n-1})\)

\(\Rightarrow ar^4=32\)

\(\Rightarrow a.2^4=32\)

\(\Rightarrow a=\frac{32}{2^4}=\frac{32}{16}=2\)

Sum of first 14 terms is \(S_{14}=\frac{a(r^{14}-1)}{r-1}\) \(\left(\because S_{n}=\frac{a(r^{n}-1)}{r-1}\right)\)

\(=\frac{2(2^{14}-1)}{2-1}\) \((\because a=2\;\&\;r=2)\)

= 2 (16384 - 1) \((\because2^{14}=16384)\)

= 32766

Correct option is A) 32, 766