Question

what is the value of 5^2+5^4+5^6........5^2n=(0.008)^-30?

Answer 1

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

Answer 2

Answer:

n=9

Step-by-step explanation:

L.H.S- 5^(2+4+6+8+……2*n)= 5^2(1+2+3+4+5…..n) = 5^(2 * {n(n+1)/2}) = 5^n(n+1).

R.H.S- (0.008)= (8/1000) = 1/125 ( on simplification) = 5^(-3)

So, (0.008)^-30 = 5^(-3 * 30) = 5^-90.

Equating L.H.S= R.H.S , we get

=> n*(n+1) = -90.

=> (n-9)(n+10) = 0. ( On factorizing)

Taking n>0 , we ignore n=-10 and we consider n= 9 as result .