Consider the sequence provided below 3^x, 3^3x, 3^5x, 3^7x,... Is the Given sequence a GP? give a reason, if it is a GP, find the 21st term

1 answer

Yes, the given sequence is a geometric progression (GP).

In a geometric progression, each term is obtained by multiplying the previous term by a common ratio. Let's find the common ratio (r) in this sequence:

To find the common ratio between consecutive terms, we divide a term by its previous term:

3^3x / 3^x = (3^3x) / (3^x) = 3^(3x - x) = 3^(2x)

So, the common ratio (r) in this GP is 3^(2x).

Now, to find the 21st term, we can use the formula for the nth term of a GP:

tn = ar^(n-1)

where tn is the nth term, a is the first term, r is the common ratio, and n is the position of the term.

In this sequence, the first term a is 3^x, the common ratio r is 3^(2x), and we want to find the 21st term, so n = 21.

t21 = (3^x) * (3^(2x))^(21-1)
= (3^x) * 3^(40x)
= 3^(x + 40x)
= 3^(41x)

Therefore, the 21st term of the given sequence is 3^(41x).