Question

5 + 15 + 45 +. ....+ 5 ( 3 ) n -1 = 5 / 2 ( 3n -1 )

Answer 1

Given :  5 + 15 + 45 +. ....+ 5 ( 3 ) ⁿ⁻¹ =  (5/2).3ⁿ⁻¹

To Find : Prove

Solution:

 5 + 15 + 45 +. ....+ 5 ( 3 ) ⁿ⁻¹

=  5 ( 1 + 3 + 9 + ...................+ 3ⁿ⁻¹)

1 + 3 + 9 + ...................+ 3ⁿ⁻¹ is a GP

a = first term = 1

r - common ratio = 3

number of terms =  n

Sum of n terms of an GP = a(rⁿ - 1)/(r - 1)

=  5 ( 1 + 3 + 9 + ...................+ 3ⁿ⁻¹)

= 5 (  3ⁿ⁻¹/(3 - 1))

= 5.3ⁿ⁻¹ / 2

= (5/2).3ⁿ⁻¹

QED

Hence proved

5 + 15 + 45 +. ....+ 5 ( 3 ) ⁿ⁻¹ =  (5/2).3ⁿ⁻¹

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