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:} \\ \frac{80}{10}=\frac{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:

$\begin{gathered} 10=s_1\cdot2^4 \\ \text{ Swap the sides:} \\ \Rightarrow s_1\cdot2^4=10 \\ \Rightarrow s_1=\frac{10}{2^4}=\frac{10}{16}=\frac{5}{8} \end{gathered}$

Answers:

s₁=5/8, r=2

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