Question

Let $ a_n $ be the $ n $-th term of an A.P. If $ S_n = a_1 + a_2 + a_3 + \cdots + a_n = 700 $, $ a_6 = 7 $, and $ S_7 = 7 $, then $ a_n $ is equal to:

• A. 56
• B. 65
• C. 64
• D. 70

Hint:

When solving problems involving A.P., use the formula for the sum and the general term to find unknowns, and simplify the system of equations to find the solution.

Answer 1

The Correct Option is C

Step 1: General Form of an A.P.
An arithmetic progression (A.P.) has the general form: \[ a_k = a_1 + (k-1)d \] where:

• \( a_1 \) is the first term,
• \( d \) is the common difference.

The sum of the first \( n \) terms (\( S_n \)) is given by: \[ S_n = \frac{n}{2} [2a_1 + (n-1)d] \] \subsection{Step 2: Use Given Information \( a_6 = 7 \)} Given the 6th term: \[ a_6 = a_1 + 5d = 7 \quad \text{(1)} \]
Step 3: Use Given Information \( S_7 = 7 \)
Given the sum of the first 7 terms: \[ S_7 = \frac{7}{2} [2a_1 + 6d] = 7 \] Simplify: \begin{align} \frac{7}{2} [2a_1 + 6d] &= 7 [2a_1 + 6d] &= 2 \quad \text{(Divide both sides by 7/2)} a_1 + 3d &= 1 \quad \text{(Divide by 2)} \quad \text{(2)} \end{align}
Step 4: Solve for \( a_1 \) and \( d \)
From equation (2): \[ a_1 = 1 - 3d \quad \text{(3)} \] Substitute (3) into equation (1): \begin{align} (1 - 3d) + 5d &= 7 1 + 2d &= 7 \\2d &= 6 \\d &= 3 \end{align} Now substitute \( d = 3 \) back into equation (3): \[ a_1 = 1 - 3(3) = -8 \]
Step 5: Find \( n \) for \( S_n = 700 \)
Using the sum formula with \( a_1 = -8 \) and \( d = 3 \): \[ S_n = \frac{n}{2} [2(-8) + (n-1)(3)] = 700 \] Simplify: \begin{align} \frac{n}{2} [-16 + 3n - 3] &= 700 \frac{n}{2} [3n - 19] &= 700 n(3n - 19) &= 1400 \\3n^2 - 19n - 1400 &= 0 \end{align} Solve the quadratic equation: \[ n = \frac{19 \pm \sqrt{(-19)^2 - 4 \cdot 3 \cdot (-1400)}}{2 \cdot 3} \] \[ n = \frac{19 \pm \sqrt{361 + 16800}}{6} \] \[ n = \frac{19 \pm \sqrt{17161}}{6} \] \[ n = \frac{19 \pm 131}{6} \] Possible solutions:

• \( n = \frac{19 + 131}{6} = 25 \)
• \( n = \frac{19 - 131}{6} \) (negative, discard)

Thus, \( n = 25 \).
Step 6: Find \( a_n \) (the \( n \)-th term)
Using the general form: \[ a_n = a_1 + (n-1)d \] For \( n = 25 \): \begin{align} a_{25} &= -8 + (25-1) \times 3 &= -8 + 72 \\&= 64 \end{align}
Conclusion
The value of \( a_n \) is \( 64 \), which corresponds to option 3.