Question

How do you find the indicated term of the geometric sequence where ${{a}_{1}}=\dfrac{1}{3},r=3,n=8$?

Answer 1

We are given that the sequence is in GP. We know that the general term of the GP is given by $a{{r}^{n-1}}$ . Now we know the values of a and r in the sequence hence we can easily find the ${{8}^{th}}$ term by substituting the values of a, r and n in the given formula.

Complete step-by-step solution:
Now let us first understand the concept of geometric sequence and arithmetic sequence.
Arithmetic sequence is a sequence in which the difference between two consecutive terms is constant. Let us say d is the common difference then the sequence is given as a, a+d, a+2d, …
Where a is the first term of the sequence. Hence the ${{n}^{th}}$ term of the sequence is given by ${{t}_{n}}=a+\left( n-1 \right)d$ .
Similarly Geometric sequence is a sequence in which the ratio of two consecutive terms is constant. Let us say that r is the common ratio between the terms and a be the first term then the sequence is given as $a,ar,a{{r}^{2}},...$ . Hence the ${{n}^{th}}$ term of the sequence is given by $a{{r}^{n-1}}$ .
Now consider the given sequence. We have the first terms as $\dfrac{1}{3}$ and r = 3.
Hence the ${{n}^{th}}$ term of the sequence is given by ${{t}_{n}}=a{{r}^{n-1}}=\dfrac{1}{3}{{.3}^{n-1}}=={{3}^{n-2}}$
Hence substituting n = 8 we get ${{t}_{8}}={{3}^{8-2}}={{3}^{6}}=729$ .
Hence the ${{8}^{th}}$ term of the sequence is 729.

Note: Now note that in the formula to find ${{n}^{th}}$ term we raise r by n – 1 and not n. The formula can be easily derived by observation. Also note that if r is not given we can easily find r as r is just the ratio of two consecutive terms. Hence $r=\dfrac{{{t}_{n+1}}}{{{t}_{n}}}$.