Question

For a given geometric sequence, the 4th term, a4, is equal to 19625, and the 9th term, a9, is equal to −95. Find the value of the 13th term? a 13 If applicable, write your answer as a fraction.

Answer 1

Answer:

The value of the $13^{th}$ term is ≈ 1.

Step-by-step explanation:

A geometric sequence is a series of numbers where each term is computed by multiplying the previous term by a constant, r also known as the common ratio.

The formula to compute the $n^{th}$ term of a GP is: $a_{n}=a_{1}\times r^{n-1}$

Here, a₁ is the first term.

It is provided that a₄ = 19625 and a₉ = 95.

Determine the value of a₁ and r as follows:

$\frac{a_{4}}{a_{9}}=\frac{a_{1}r^{4-1}}{a_{1}r^{9-1}} \\\frac{19625}{95}= \frac{r^{3}}{r^{8}}r^{5}=\frac{95}{19625}\\ r=(\frac{95}{19625})^{1/5}\\=0.344$

The common ratio is, r = 0.344.

The value of a₁ is:

$a_{4}=19625\\a_{1}\times(0.344)^{3}=19625\\a_{1}=\frac{19625}{0.040707584} \\=482096.898\\\approx482097$

The first term is, a₁ = 482097.

13th term of this geometric sequence is:

$a_{13}=a_{1}\times r^{13-1}\\=482097\times (0.344)^{12}\\=1.3234\\\approx1$

Thus, the $13^{th}$ term is approximately equal to 1.