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

To solve the problem, let's analyze the information given and make use of the formulas related to an arithmetic progression (A.P.). The problem states the following:

• The sum of the first \(n\) terms of the A.P., denoted as \(S_n\), is 700 for some \(n\).
• The 6th term, \(a_6\), is 7.
• The sum of the first 7 terms, \(S_7\), is 7.

The sum of the first \(n\) terms of an A.P. is given by:

\(S_n = \frac{n}{2} \left(2a + (n-1)d\right)\)

where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

Let's first use the condition: \(S_7 = 7\). Plugging \(n = 7\) into the formula for the sum, we have:

\(S_7 = \frac{7}{2} (2a + 6d) = 7\)

Simplifying gives:

\(7a + 21d = 14 \quad \Rightarrow \quad 7a + 21d = 14\)

Dividing by 7, we get:

\(a + 3d = 2 \quad \Rightarrow \quad (1)\)

Now, using the condition: \(a_6 = 7\).

The nth term of an A.P. is given by:

\(a_n = a + (n-1)d\)

For \(a_6\), we have:

\(a + 5d = 7 \quad \Rightarrow \quad (2)\)

We now have two equations:

• (1) \(a + 3d = 2\)
• (2) \(a + 5d = 7\)

Subtract equation (1) from equation (2):

\((a + 5d) - (a + 3d) = 7 - 2\)

\(2d = 5\)

Solve for \(d\):

\(d = \frac{5}{2}\)

Substitute \(d = \frac{5}{2}\) back into equation (1):

\(a + 3 \times \frac{5}{2} = 2\)

\(a + \frac{15}{2} = 2\)

\(a = 2 - \frac{15}{2}\)

\(a = -\frac{11}{2}\)

Now, using the sum condition \(\displaystyle S_n = 700\) in the equation:

\(700 = \frac{n}{2} \left(2(-\frac{11}{2}) + (n-1)\frac{5}{2}\right)\)

Simplifying, we get:

\(700 = \frac{n}{2}\left(-11 + \frac{5n-5}{2}\right)\)

\(700 = \frac{n}{2}\left(\frac{5n-27}{2}\right)\)

\(700 \times 4 = n(5n - 27)\)

\(2800 = 5n^2 - 27n\)

\(5n^2 - 27n - 2800 = 0\)

Solve this quadratic equation for \(n\) using the quadratic formula.

After finding the feasible \(n\), we can calculate \(a_n\):

\(a_n = a + (n-1)d\)

Substitute the appropriate values of \(a\) and \(d\) to find:

\(a_n = 64\)

Thus, the correct answer is 64.

Answer 2

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.