Chapter 9, Problem 27MCQ

Solution

To write the first four terms of an arithmetic sequence with a common difference of 3 and a third term of 10. Then write a recursive definition and an explicit formula for the sequence.

The first four terms of the sequence are 4, 7, 10 and 13. The recursive definition for the sequence is $a_n=a_{n-1}+3$ where $a_1=4$ , and the explicit formula is $a_n=3n+1$ .

Given information:

An arithmetic sequence with a common difference of 3 and a third term of 10.

Calculation:

Given, $d=3$ and $a_3=10$ .

Substitute $n=3$ in the formula $a_n=a_1+\left(n-1\right)d$ and the substitute the respective values to find the first term.

  $\begin{array}{c}{a}_3=a_1+\left(3-1\right)d\\ 10=a_1+2\cdot 3\\ 10=a_1+6\\ a_1=4\end{array}$

Now, find the second and the fourth terms.

  $\begin{array}{c}{a}_2=a_1+\left(2-1\right)d=4+1\cdot 3=7\\ a_4=a_1+\left(4-1\right)d=4+3\cdot 3=13\end{array}$

Since the first term of the sequence is $a_1=4$ and the common difference is $d=3$ , the recursive definition can be written as $a_n=a_{n-1}+3$ where $a_1=4$ .

Use $a_1=4$ and $d=3$ in the formula $a_n=a_1+\left(n-1\right)d$ to find the explicit formula.

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =4+\left(n-1\right)\cdot 3\\ =3n+1\end{array}$

So, the explicit formula is $a_n=3n+1$ .

Conclusion:

The first four terms of the sequence are 4, 7, 10 and 13. The recursive definition for the sequence is $a_n=a_{n-1}+3$ where $a_1=4$ , and the explicit formula is $a_n=3n+1$ .