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[tex]_{n}[/tex] = a[tex]r^{n-1}[/tex], where
The sum of the nth term is S[tex]_{n}[/tex] = [tex]\frac{a(1-r^{n})}{1-r}[/tex]
∵ The second term of a geometric sequence is 14.5
∴ n = 2
∴ a[tex]_{2}[/tex] = 14.5
∵ a[tex]_{2}[/tex] = ar
→ Equate the right sides of a[tex]_{2}[/tex] by 14.5
∵ ar = 14.5 ⇒ (1)
∵ The fifth term is 1.8125
∴ n = 5
∴ a[tex]_{5}[/tex] = 1.8125
∵ a[tex]_{5}[/tex] = a[tex]r^{4}[/tex]
→ Equate the right sides of a[tex]_{5}[/tex] by 14.5
∵ a[tex]r^{4}[/tex] = 1.8125 ⇒ (2)
→ Divide equation (2) by equation (1)
∵ [tex]\frac{ar^{4}}{ar}[/tex] = [tex]\frac{1.8125}{14.5}[/tex]
∴ 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[tex]_{5}[/tex] = [tex]\frac{29[1-[0.5]^{5})}{1-0.5}[/tex]
∴ S[tex]_{5}[/tex] = 56.1875
c) The sum of the first 5 terms is 56.1875
