Question

Find the sum to 5 terms of the geometric progression whose first term is 54 and fourth term is 2

Answer 1

nth term of a geometric:

an = a1 rⁿ ⁻ ¹

a1 = initial value

r = common ratio

In this case:

a1 = 54

a4 = 2

a4 = a1 r⁴ ⁻ ¹ = 2

54 ∙ r³ = 2

r³ = 2 / 54

r³ = 2 / 2 ∙ 27

r³ = 1 / 27

r = ∛ (1 / 27 )

r = ∛1 / ∛27

r = 1 / 3

Sn = Sum first n terms of GP

Sn = a1 ( 1 - rⁿ ) / ( 1 - r )

S5 = Sum first 5 terms of GP

S5 = a1 ( 1 - r⁵ ) / ( 1 - r )

S5 = 54 [ 1 - ( 1 / 3 )⁵ ] / ( 1 - 1 / 3 )

S5 = 54 [ 1 - ( 1⁵ / 3⁵ ) ] / ( 3 / 3 - 1 / 3 )

S5 = 54 ( 1 - 1 / 243 ) / ( 2 / 3 )

S5 = 54 ( 243 / 243 - 1 / 243 ) / ( 2 / 3 )

S5 = 54 ( 242 / 243 ) / ( 2 / 3 )

S5 = ( 54 ∙ 242 / 243 ) / ( 2 / 3 )

S5 = 54 ∙ 242 ∙ 3 / 243 ∙ 2

S5 = 39204‬ / 486

S5 = 162 ∙ 242‬ / 162 ∙ 3

‬S5 = 242‬ / 3

Proof:

a1 = 54

a2 = 54 / 3 = 18

a3 = 18 / 3 = 6

a4 = 6 / 3 = 2

a5 = 2 / 3

a1 + a2 + a3 + a4 + a5 = 54 + 18 + 6 + 2 + 2 / 3 =

80 + 2 / 3 = 240 / 3 + 2 / 3 = 242 / 3