Question

Find n in the equation 5^2×5^4×5^6.........5^2n=(125)^30

Answer 1

Hi ,

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By Exponential laws :

1 ) a^m × a^n = a^m+n

2 ) If a^m = a^n then m = n

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5² × 5⁴ × 5^6 × ....× 5^2n = ( 125 )^30

5^2+4+6+...+2n = ( 5³ )^30

5^2+4+6+...+2n = 5^90

Therefore ,

2 + 4 + 6 + ...+ 2n = 90---( 1 )

2 , 4 , 6 ,. .. 2n are in A.P

first term = a = 2

common difference = a2 - a1

d = 4 - 2 = 2

last term = nth term = 2n

Sum of n terms in an A.P = 90 [ from( 1 ) ]

( n/2 ) [ 2 + 2n ] = 90

( 2n/2 ) ( 1 + n ) = 90

n( 1 + n ) - 90 = 0

n² + n - 90 = 0

n² - 9n + 10n - 90 = 0

n( n - 9 ) + 10 ( n - 9 ) = 0

( n - 9 ) ( n + 10 ) = 0

n - 9 = 0 or n + 10 = 0

n = 9 or n = - 10

I hope this helps you.

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