Answer :
Answer:
The value of the [tex]13^{th}[/tex] 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 [tex]n^{th}[/tex] term of a GP is: [tex]a_{n}=a_{1}\times r^{n-1}[/tex]
Here, a₁ is the first term.
It is provided that a₄ = 19625 and a₉ = 95.
Determine the value of a₁ and r as follows:
[tex]\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[/tex]
The common ratio is, r = 0.344.
The value of a₁ is:
[tex]a_{4}=19625\\a_{1}\times(0.344)^{3}=19625\\a_{1}=\frac{19625}{0.040707584} \\=482096.898\\\approx482097[/tex]
The first term is, a₁ = 482097.
13th term of this geometric sequence is:
[tex]a_{13}=a_{1}\times r^{13-1}\\=482097\times (0.344)^{12}\\=1.3234\\\approx1[/tex]
Thus, the [tex]13^{th}[/tex] term is approximately equal to 1.