Question

Find the values of S1 and R for a geometric sequence with S5=10 and S8=80

Answer 1

The 5th term and 8th term of a geometric sequence are given to be 10 and 80, respectively.

It is required to find the first term and the common ratio.

Recall the Explicit Formula for a geometric sequence:

`s_n=s_(1\cdot)r^(n-1)`

Substitute n=5 into the formula:

`\begin{gathered} s_5=s_1\cdot r^(5-1) \\ \Rightarrow s_5=s_1\cdot r^4 \end{gathered}`

Substitute s₅=10 into the equation:

`10=s_1\cdot r^4`

Use the same procedure for s₈ to get the second equation:

`80=s_1\cdot r^7`
`\begin{gathered} \text{ Divide the second equation by the first equation:} \\ (80)/(10)=(s_1\cdot r^7)/(s_1\cdot r^4) \end{gathered}`

Simplify the equation and solve for r:

`\begin{gathered} 8=r^(7-4) \\ \Rightarrow8=r^3 \\ \Rightarrow r=\sqrt[3]{8}=2 \end{gathered}`

Substitute r=2 into the first equation:

`\begin{gathered} 10=s_1\cdot2^4 \\ \text{ Swap the sides:} \\ \Rightarrow s_1\cdot2^4=10 \\ \Rightarrow s_1=(10)/(2^4)=(10)/(16)=(5)/(8) \end{gathered}`

Answers:

s₁=5/8, r=2

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