Question

The product of the first 11 terms in a G.P is 2048. Find the 6thterm?​

Answer 1

Answer:

The 6th term of the GP is 2

Step-by-step explanation:

Let the first term of the GP be a and common ratio be r

Then the GP upto 11 terms will be

a, ar, ar^2, ar^3, ....., ar^10

The product of the first 11 terms

$2048=a\times ar\times ar^2\times ar^3\times ........\times ar^{10}$

$\implies 2048=a^{11}r^ {(1+2+3+...+10)}$

$\implies 2048=a^{11}r^{55}$

$\implies 2048=(ar^5)^{11}$

$\implies ar^5= (2048)^{1/11}=(2^{ 11})^{1/11}$

$\implies ar^5=2$

But ar^5 is nothing but the 6th term of the GP

Therefore, the 6th term of the GP is 2

Answer 2

The product of the first 11 terms in a geometric progression is 2048. Find the 6th term?

solution : let first 11 terms in geometric progression are ; a/r^5, a/r^4, a/r^3, a/r^2, a/r, a , ar, ar^2, a^3, ar^4, ar^5.

a/c to question,

product of 11 terms = 2048

⇒ (a/r^5 × a/r^4 × a/r^3 × a/r^2 × a/r × a × ar × ar^2 × ar^3 × ar^4 × ar^5) = 2048

⇒ a¹¹ = 2048 = (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 )

⇒a¹¹ = 2¹¹

⇒ a = 2

now see series here 6th term is a.

so, 6th term of given geometric progression , a = 2.