Question

Given the following information about the arithmetic sequence an, find a17.
a3=13
a13=43

Answer 1

`$ \textbf{a}_{\textbf{17}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{55} $`

Explanation:

The
`$ n^(th) $` term of an arithmetic sequence is given by:

`$ \textbf{a}_{\textbf{n}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{a} \hspace{1mm} \textbf{+} \hspace{1mm} \textbf{(n - 1)d} $`

where a is the first term of the sequence

and d is the common difference.

We are given the
`$ 3^(rd) $` and the
`$ 13^(th) $` term of the sequence.

We are asked to find the
`$ 17^(th) $` term.

From the formula, we can write

`$ a_3 = a + (3 - 1)d $`

`$ \implies 13 = a + 2d \hspace{6mm} \hdots (1) $`

Also,
`$ a_(13) = a + (13 - 1)d $`

`$ \implies 43 = a + 12d \hspace{6mm} \hdots (2) $`

Now, we solve Equation (1) and (2) for a and d.

Solving we get:

a = 7; d = 3

Therefore,
`$ 17^(th) $` term,
`$ a_(17) $` can now be calculated.

`$ a_(17) = a + (17 - 1)d $`

`$ \implies a_(17) = 7 + 16(3) $`

`$ \implies \textbf{a}_{\textbf{17}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{55} $`

Therefore, the
`$ 17^(th) $` term of the sequence is 55.

Hence, the answer.