Question

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4 , then the sum of its first twelve terms is

• A. -20
• B. -24
• C. -26
• D. -22

Hint:

When working with A.P. sums, you can simplify equations before solving. Notice how \(S_n = \dots\) directly translates to a linear equation in \(a\) and \(d\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the sum of the first 4 terms and the first 6 terms of an Arithmetic Progression (A.P.). We need to find the first term (\(a\)) and the common difference (\(d\)) to calculate the sum of the first 12 terms.


Step 2: Key Formula or Approach:
The sum of the first \(n\) terms of an A.P. is given by: \[ S_n = \frac{n}{2} [2a + (n-1)d] \] We will set up two linear equations using the given data and solve for \(a\) and \(d\).


Step 3: Detailed Explanation:
Given \(S_4 = 6\): \[ \frac{4}{2} [2a + (4-1)d] = 6 \] \[ 2 [2a + 3d] = 6 \] \[ 2a + 3d = 3 \quad \dots \text{(Equation 1)} \] Given \(S_6 = 4\): \[ \frac{6}{2} [2a + (6-1)d] = 4 \] \[ 3 [2a + 5d] = 4 \] \[ 6a + 15d = 4 \quad \dots \text{(Equation 2)} \] To solve these equations, multiply Equation 1 by 3: \[ 6a + 9d = 9 \quad \dots \text{(Equation 3)} \] Subtract Equation 3 from Equation 2: \[ (6a + 15d) - (6a + 9d) = 4 - 9 \] \[ 6d = -5 \implies d = -\frac{5}{6} \] Substitute the value of \(d\) into Equation 1: \[ 2a + 3\left(-\frac{5}{6}\right) = 3 \] \[ 2a - \frac{5}{2} = 3 \] \[ 2a = 3 + \frac{5}{2} = \frac{11}{2} \implies a = \frac{11}{4} \] Now, we find the sum of the first 12 terms (\(S_{12}\)): \[ S_{12} = \frac{12}{2} [2a + (12-1)d] \] \[ S_{12} = 6 [2a + 11d] \] Substitute \(a = 11/4\) and \(d = -5/6\): \[ S_{12} = 6 \left[ 2\left(\frac{11}{4}\right) + 11\left(-\frac{5}{6}\right) \right] \] \[ S_{12} = 6 \left[ \frac{11}{2} - \frac{55}{6} \right] \] Find a common denominator for the terms inside the bracket: \[ S_{12} = 6 \left[ \frac{33 - 55}{6} \right] \] \[ S_{12} = 33 - 55 = -22 \]

Step 4: Final Answer:
The sum of the first twelve terms is -22.