Question

Complete the statement for each arithmetic sequence.

-14 is the ___?__ term of $2\frac{1}{5},2,1\frac{4}{5},\mathrm{...}$.

Short answer

-14 is the82nd termof the arithmetic sequence $2\frac{1}{5},2,1\frac{4}{5},\mathrm{...}$.

Step-by-step solution

Step 1. Given Information.

Given arithmetic sequence is $2\frac{1}{5},2,1\frac{4}{5},\mathrm{...}$

Step 2. Calculation.

The n^th term of an arithmetic sequence is given by $a_n=a_1+(n-1)d$.

Here the first term is $a_1=2\frac{1}{5}$

The difference between the first two terms is $2-2\frac{1}{5}=-\frac{1}{5}$

The difference between the second two terms is $1\frac{4}{5}-2=-\frac{1}{5}$

The common difference is $d=-\frac{1}{5}$

The given nth term is $a_n=-14$

Plugging the values in the formula:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ -14=2\frac{1}{5}+(n-1)(-\frac{1}{5})\\ -14-2\frac{1}{5}=(n-1)(-\frac{1}{5})\\ (n-1)(-\frac{1}{5})=-\frac{81}{5}\\ n-1=81\\ n=81+1\\ n=82\end{array}$

Step 3. Conclusion.

Hence, -14 is the 82nd term of the arithmetic sequence $2\frac{1}{5},2,1\frac{4}{5},\mathrm{...}$.