Question

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

Answer 1

Answer:

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

Step-by-step 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$_{n}$ = $\frac{a(1-r^{n})}{1-r}$

∵ 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)

∵ $\frac{ar^{4}}{ar}$ = $\frac{1.8125}{14.5}$

∴ 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$_{5}$ = $\frac{29[1-[0.5]^{5})}{1-0.5}$

∴ S$_{5}$ = 56.1875

c) The sum of the first 5 terms is 56.1875