Question

5^2×5^4×5^6×........ ×5^2n==(0.008)^-30 find value of n?​

Answer 1

Answer: 9

Step-by-step explanation:

The question can be written as

5^(2+4+6+….2n)=(.0008)^-30

(0.0008)^-30=(125)^30=5^(90)

5^(2+4+6+…+2n)=5^(90)

equating the powers,

2+4+6+…+2n=90

it is an arithmetic progression.

the formula for sum of an A.P is (n/2)(2a+(n-1)d).

where n is the number of elements, a is the first value and d is the difference.

in this A.P, a=2,d=2.

(n/2)(2a+(n-1)d)=90.

n(4+(n-1)2)=180

2n^2+2n-180=0

solving the equation, n=9

the last element is a+(n-1)d=2+(8)2=18

since in the question it is given as 2n,n=18/2=9