Sure, let's solve the problem step-by-step.
Given the terms [tex]\(x+6\)[/tex], [tex]\(3x\)[/tex], and [tex]\(2x+9\)[/tex] which are in Arithmetic Progression (A.P.), we need to determine the value of [tex]\(x\)[/tex] and the next three terms of the progression.
1. Understanding Arithmetic Progression:
- For three terms to be in an arithmetic progression, the difference between consecutive terms must be constant.
- This can be expressed as:
[tex]\[(\text{2nd term} - \text{1st term}) = (\text{3rd term} - \text{2nd term})\][/tex]
2. Set up the equation:
Using the given terms [tex]\(x+6\)[/tex], [tex]\(3x\)[/tex], and [tex]\(2x+9\)[/tex], we set up the equation based on the A.P. property:
[tex]\[3x - (x+6) = (2x+9) - 3x\][/tex]
3. Simplify the equation:
[tex]\[3x - x - 6 = 2x + 9 - 3x\][/tex]
[tex]\[2x - 6 = 9 - x\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Add [tex]\(x\)[/tex] to both sides:
[tex]\[2x + x - 6 = 9\][/tex]
[tex]\[3x - 6 = 9\][/tex]
Add 6 to both sides:
[tex]\[3x = 15\][/tex]
Divide by 3:
[tex]\[x = 5\][/tex]
5. Find the first three terms:
Substituting [tex]\(x = 5\)[/tex] back into the expressions for the terms:
[tex]\[
\begin{align*}
\text{First term} \ (T_1) &= x + 6 = 5 + 6 = 11, \\
\text{Second term} \ (T_2) &= 3x = 3 \times 5 = 15, \\
\text{Third term} \ (T_3) &= 2x + 9 = 2 \times 5 + 9 = 19.
\end{align*}
\][/tex]
6. Determine the common difference:
[tex]\[
\text{Common difference} \ d = T_2 - T_1 = 15 - 11 = 4.
\][/tex]
7. Calculate the next three terms:
Using the common difference:
[tex]\[
\begin{align*}
\text{Fourth term} \ (T_4) &= T_3 + d = 19 + 4 = 23, \\
\text{Fifth term} \ (T_5) &= T_4 + d = 23 + 4 = 27, \\
\text{Sixth term} \ (T_6) &= T_5 + d = 27 + 4 = 31.
\end{align*}
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is [tex]\(5\)[/tex] and the progression is [tex]\(11, 15, 19, 23, 27, 31\)[/tex].
