Certainly! Let's solve the given equation step by step:
[tex]\[
\frac{a + 5d}{a} = \frac{a + 11d}{a + 5d}
\][/tex]
Step 1: Cross-multiply to eliminate the fractions.
[tex]\[
(a + 5d)(a + 5d) = a(a + 11d)
\][/tex]
Step 2: Expand both sides of the equation.
[tex]\[
(a + 5d)(a + 5d) = a^2 + 10ad + 25d^2
\][/tex]
[tex]\[
a(a + 11d) = a^2 + 11ad
\][/tex]
Step 3: Set the expanded forms equal to each other.
[tex]\[
a^2 + 10ad + 25d^2 = a^2 + 11ad
\][/tex]
Step 4: Subtract [tex]\(a^2\)[/tex] from both sides to simplify.
[tex]\[
10ad + 25d^2 = 11ad
\][/tex]
Step 5: Subtract [tex]\(10ad\)[/tex] from both sides to further simplify.
[tex]\[
25d^2 = ad
\][/tex]
Step 6: Divide both sides by [tex]\(d\)[/tex] (assuming [tex]\(d \neq 0\)[/tex]) to isolate [tex]\(a\)[/tex].
[tex]\[
25d = a
\][/tex]
So the solution for [tex]\(a\)[/tex] in terms of [tex]\(d\)[/tex] is:
[tex]\[
a = 25d
\][/tex]
Therefore, the value of [tex]\(a\)[/tex] that satisfies the original equation is:
[tex]\[
a = 25d
\][/tex]
