Question

The terms \(x_5 = -4\), \(x_1, x_2, \dots, x_{100}\) are in an arithmetic progression (AP). It is also given that \(2x_6 + 2x_9 = x_{11} + x_{13}\). Find \(x_{100}\).

• A. -194
• B. 206
• C. 204
• D. -196

Answer 1

The Correct Option is A

Step 1: General formula for terms in an AP
The general term of an AP is:
\[x_n = x_1 + (n-1)d,\]
where:
\(x_1\) is the first term, and
\(d\) is the common difference.
Substituting \(n = 5\), we get:
\[x_5 = x_1 + 4d = -4. \quad \text{(Equation 1)}\]
Step 2: Rewrite the given condition in terms of \(x_1\) and \(d\)
The terms \(x_6, x_9, x_{11}, x_{13}\) in the AP are:
\[x_6 = x_1 + 5d, \quad x_9 = x_1 + 8d, \quad x_{11} = x_1 + 10d, \quad x_{13} = x_1 + 12d.\]
The given condition is:
\[2x_6 + 2x_9 = x_{11} + x_{13}.\]
Substitute the expressions for the terms:
\[2(x_1 + 5d) + 2(x_1 + 8d) = (x_1 + 10d) + (x_1 + 12d).\]
Simplify:
\[2x_1 + 10d + 2x_1 + 16d = x_1 + 10d + x_1 + 12d.\]
Combine terms:
\[4x_1 + 26d = 2x_1 + 22d.\]
Simplify further:
\[2x_1 + 4d = 0. \quad \text{(Equation 2)}\]
Step 3: Solve for \(x_1\) and \(d\)
From Equation 2:
\[x_1 = -2d.\]
Substitute \(x_1 = -2d\) into Equation 1:
\[-2d + 4d = -4.\]
Simplify:
\[2d = -4 \implies d = -2.\]
Now substitute \(d = -2\) into \(x_1 = -2d\):
\[x_1 = -2(-2) = 4.\]
Step 4: Find \(x_{100}\)
The general term of the AP is:
\[x_n = x_1 + (n-1)d.\]
Substitute \(n = 100\), \(x_1 = 4\), and \(d = -2\):
\[x_{100} = 4 + (100-1)(-2).\]
Simplify:
\[x_{100} = 4 + 99(-2),\]
\[x_{100} = 4 - 198,\]
\[x_{100} = -194.\]
Final Answer
The 100th term of the AP is:
\[\boxed{-194}.\]