Chapter 9.4, Problem 14PPSE

Solution

a.

To find the score of the student in the sixth grade.

The score in the sixth test is $95$ .

Given information:

The student scores 75, 79, and 83 in the first three tests.

Calculation:

Observe that the improvement in scores represent an arithmetic sequence with first term $a_1=75$ and common difference $d=4$ .

Find the nth term of the sequence.

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =75+\left(n-1\right)4\\ =4n+71\end{array}$

Now, find the score in the sixth test.

  $\begin{array}{c}{a}_6=4\cdot 6+71\\ =24+71\\ =95\end{array}$

Therefore, the score in the sixth test is $95$ .

b.

To find the total score for all six tests.

The total score of the student for all the six tests is 510.

Given information:

The student scores 75, 79, and 83 in the first three tests.

Calculation:

Since the common difference is 4, all the six terms of the sequence can be written as:

  $75,79,83,87,91,95$

Now, add all the terms.

  $\begin{array}{c}75+79+83+87+91+95=\frac{6}{2}\left(75+95\right)\\ =3\left(170\right)\\ =510\end{array}$

Therefore, the total score of the student for all the six tests is 510.