Question

Review
If the sixth term of a GP be 2, then the product of first
11 terms is​

Answer 1

Answer:

2048

Step-by-step explanation:

Given,

6th term of a G.P = 2

To Find :-

Product of first 11 terms.

How To Do :-

First we by using general term of an A.P formula we need to equate the 6th term of an A.P. Then we will get an equation. After that we need to find the product of 11terms of G.P. After simplifying it we can see that there is chance to use the before equation in that. SO by putting the value of before equation. We will get the value of product of first 11 terms.

Formula Required :-

General term of a G.P :-

$a_n=a\times r^{n-1}$

Solution :-

$a_6=a\times r^{6-1}$

$2=ar^5$

[Let it be equation - 1]

Product of 11 terms of a G.P :-

Let , the 11 terms of a G.P be :-

a , ar , ar² , ar³ , ar⁴ , ar⁵ , ar⁶ , ar⁷ , ar⁸ , ar⁹ , ar¹⁰

Product of the terms :-

= a × ar × ar² × ar³ × ar⁴ × ar⁵ × ar⁶ × ar⁷ × ar⁸ × ar⁹ × ar¹⁰

Multiplying 'a' terms together and 'r' terms together :-

= (a × a  ×  a  ×  a  ×  a  ×  a  ×  a  ×  a  ×  a  ×  a  ×  a)  ×  (r  ×  r²  ×  r³  ×  r⁴  ×  r⁵  ×  r⁶  ×  r⁷  ×  r⁸  ×  r⁹  ×  r¹⁰)

$=(a^{1+1+1+1+1+1+1+1+1+1+1})\times (r^{1+2+3+4+5+6+7+8+9+10})$

$=a^{11}\times r^{55}$

= (a¹)¹¹ × (r⁵)¹¹

= (a × r⁵ ) ¹¹

Substituting Value of Equation - 1 :-

= 2¹¹

= 2048

∴ Product of first 11 terms of a G.P = 2048.