Question

For a given geometric sequence, the 8th term is equal to 23/625, and the 13th term is 115. What is the 16th term

Answer 1

14,375

Explanation :

Firstly we'll find the common ratio (r) of the sequence

• 23/625(r)^(13-8) = 115
• 23/625(r)^5 = 115
• (r)^5 = (115*625)/23
• (r)^5 = 3,125
• r= ⁵√3,125
• r = 5

now, we'll use the recursive formula to find the 16th term(a16)

• a16 = a15(r) = a14(r)(r) = a13(r)(r)(r)
• a16 = a13(r)^3
• a(16) = 115(5)^3
• a(16) = 14,375

Thus, the 16th term is 14,375.

Answer 2

a₁₆ = 14375

Explanation:

The general form of the nth term a geometric sequence is given by:

`\large\boxed{a_n=ar^(n-1)}`

where:

• 
  `a_n` is the nth term.
• a is the first term.
• r is the common ratio.
• n is the position of the term.

Given that the 8th term is equal to 23/625, and the 13th term is 115:

`a_8=(23)/(625)\implies ar^(7)=(23)/(625)`

`a_(13)=115\implies ar^(12)=115`

Divide the 13th term by the 8th term to eliminate a:

`\begin{aligned}(a_(13))/(a_8)=(ar^(12))/(ar^(7))&=(115)/((23)/(625))\\\\(r^(12))/(r^(7))&=115 \cdot (625)/(23)\\\\r^(12-7)&=3125\\\\r^(5)&=3125\end{aligned}`

Solve for r:

`\begin{aligned}r&=\sqrt[5]{3125}\\\\r&=5\end{aligned}`

Now, substitute the found value of r into one of the term equations and solve for a. Let's use a₁₃:

`\begin{aligned}a(5)^(12)&=115\\\\a&=(115)/((5)^(12))\\\\a&=(115)/(244140625)\end{aligned}`

Therefore, the formula for the nth term of the given geometric sequence is:

`\large\boxed{a_n=\left((115)/(244140625)\right)5^(n-1)}`

To find the 16th term, simply substitute n = 16 into the formula:

`a_(16)=\left((115)/(244140625)\right)5^(16-1)`

`a_(16)=\left((115)/(244140625)\right)5^(15)`

`a_(16)=\left((115)/(244140625)\right)30517578125`

`a_(16)=(3509521484375)/(244140625)`

`a_(16)=14375`

Therefore, the 16th term of the given geometric sequence is 14375.