Question

20 The second term of a geometric sequence is 14.5 and the fifth

term is 1.8125
a Determine the common ratio,
b Find the value of the first term.
€ Find the sum of the first 5 terms,

Answer 1

a. The common ratio is 0.5

b) The value of the first term is 29

c) The sum of the first 5 terms is 56.1875

Explanation:

The nth term of the geometric sequence is a
`_(n)` = a
`r^(n-1)`, where

• a is the 1st term

• r is the common ratio

The sum of the nth term is S
`_(n)` =
`(a(1-r^(n)))/(1-r)`

∵ The second term of a geometric sequence is 14.5

∴ n = 2

∴ a
`_(2)` = 14.5

∵ a
`_(2)` = ar

→ Equate the right sides of a
`_(2)` by 14.5

∵ ar = 14.5 ⇒ (1)

∵ The fifth term is 1.8125

∴ n = 5

∴ a
`_(5)` = 1.8125

∵ a
`_(5)` = a
`r^(4)`

→ Equate the right sides of a
`_(5)` by 14.5

∵ a
`r^(4)` = 1.8125 ⇒ (2)

→ Divide equation (2) by equation (1)

∵
`(ar^(4))/(ar)` =
`(1.8125)/(14.5)`

∴ r³ = 0.125

→ Take ∛ for both sides

∴ r = 0.5

a. The common ratio is 0.5

→ Substitute the value of r in equation (1) to find a

∵ a(0.5) = 14.5

∴ 0.5a = 14.5

→ Divide both sides by 0.5

∴ a = 29

b) The value of the first term is 29

∵ n = 5

∴ S
`_(5)` =
`(29[1-[0.5]^(5)))/(1-0.5)`

∴ S
`_(5)` = 56.1875

c) The sum of the first 5 terms is 56.1875