Answer :

Sure! Let's solve this step-by-step:

1. Identifying the Given Information:
- The first term ([tex]\(a_1\)[/tex]) of the arithmetic progression (AP) is [tex]\(-8\)[/tex].
- The ratio of the 9th term ([tex]\(a_9\)[/tex]) to the first term ([tex]\(a_1\)[/tex]) is [tex]\(5:8\)[/tex].

2. Understanding the Formula for the nth Term of an AP:
- The nth term ([tex]\(a_n\)[/tex]) of an AP can be calculated using the formula:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.

3. Setting Up the Equation for the 9th Term (a_9):
- The 9th term [tex]\(a_9\)[/tex] can be written as:
[tex]\[
a_9 = a_1 + 8 \cdot d
\][/tex]
- Substituting [tex]\(a_1 = -8\)[/tex]:
[tex]\[
a_9 = -8 + 8d
\][/tex]

4. Using the Given Ratio:
- According to the problem, the ratio of [tex]\(a_9\)[/tex] to [tex]\(a_1\)[/tex] is [tex]\(5:8\)[/tex]. Therefore:
[tex]\[
\frac{a_9}{a_1} = \frac{5}{8}
\][/tex]
- Substituting [tex]\(a_1 = -8\)[/tex] and the equation for [tex]\(a_9\)[/tex]:
[tex]\[
\frac{-8 + 8d}{-8} = \frac{5}{8}
\][/tex]

5. Solving for the Common Difference (d):
- Simplify the equation:
[tex]\[
\frac{-8 + 8d}{-8} = \frac{5}{8}
\][/tex]
- Multiply both sides by [tex]\(-8\)[/tex]:
[tex]\[
-8 + 8d = -5
\][/tex]
- Add 8 to both sides of the equation:
[tex]\[
8d = 3
\][/tex]
- Divide both sides by 8:
[tex]\[
d = \frac{3}{8}
\][/tex]
- Therefore, the common difference [tex]\(d\)[/tex] is [tex]\(\frac{3}{8} = 0.375\)[/tex].

So, the common difference [tex]\(d\)[/tex] is [tex]\(0.375\)[/tex].

Other Questions