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a) What is the common difference of an arithmetic progression (AP) given [tex]$a_{21} - a_7 = 84$[/tex]?

b) Which term of the AP [tex]$3, 8, 13, \ldots$[/tex] is 78?

Answer :

Sure! Let's go through the solution step-by-step:

### Part a) Finding the Common Difference

We are given that in an arithmetic progression (AP), the difference between the 21st term and the 7th term is 84.

1. Formula for the nth term of an AP:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, and [tex]\(d\)[/tex] is the common difference.

2. Write the expressions for the 21st and 7th terms:
[tex]\[
a_{21} = a_1 + 20d
\][/tex]
[tex]\[
a_7 = a_1 + 6d
\][/tex]

3. Calculate the difference:
[tex]\[
a_{21} - a_7 = (a_1 + 20d) - (a_1 + 6d)
\][/tex]
[tex]\[
a_{21} - a_7 = 14d
\][/tex]

4. Set the difference equal to 84:
[tex]\[
14d = 84
\][/tex]

5. Solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{84}{14} = 6
\][/tex]

So, the common difference [tex]\(d\)[/tex] is 6.

### Part b) Finding Which Term is 78

We have the AP: 3, 8, 13, ...

1. Identify the first term and the common difference:
- First term ([tex]\(a_1\)[/tex]) is 3.
- The common difference ([tex]\(d\)[/tex]) is 5, found by subtracting the first term from the second term: [tex]\(8 - 3 = 5\)[/tex].

2. Use the formula to find the nth term equal to 78:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
Substitute the known values:
[tex]\[
78 = 3 + (n-1) \cdot 5
\][/tex]

3. Solve for [tex]\(n\)[/tex]:
[tex]\[
78 - 3 = (n-1) \cdot 5
\][/tex]
[tex]\[
75 = (n-1) \cdot 5
\][/tex]
[tex]\[
n-1 = \frac{75}{5}
\][/tex]
[tex]\[
n-1 = 15
\][/tex]
[tex]\[
n = 16
\][/tex]

The term in the sequence that is 78 is the 16th term.

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