Question

If the sum of first 4 terms of an AP is 6 and sum of first 6 terms is 4, then sum of first 12 terms of AP is :

• A. -22
• B. -20
• C. 22
• D. 20

Hint:

When given sums of different numbers of terms in an AP, you can set up a system of linear equations for the first term 'a' and common difference 'd'. Solving this system is the key to finding any other property of the AP.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the sum of the first 4 terms (S\(_4\)) and the sum of the first 6 terms (S\(_6\)) of an arithmetic progression (AP). We need to find the sum of the first 12 terms (S\(_{12}\)).

Step 2: Key Formula:
The sum of the first n terms of an AP is given by the formula: \[ S_n = \frac{n}{2}[2a + (n-1)d] \] where 'a' is the first term and 'd' is the common difference.

Step 3: Setting up and Solving Equations:
We are given two conditions: 1. S\(_4\) = 6 \[ \frac{4}{2}[2a + (4-1)d] = 6 \Rightarrow 2(2a + 3d) = 6 \Rightarrow 2a + 3d = 3 \quad \text{(Equation 1)} \] 2. S\(_6\) = 4 \[ \frac{6}{2}[2a + (6-1)d] = 4 \Rightarrow 3(2a + 5d) = 4 \Rightarrow 2a + 5d = \frac{4}{3} \quad \text{(Equation 2)} \] Now, we solve this system of linear equations. Subtract Equation 1 from Equation 2: \[ (2a + 5d) - (2a + 3d) = \frac{4}{3} - 3 \] \[ 2d = \frac{4-9}{3} = -\frac{5}{3} \Rightarrow d = -\frac{5}{6} \] Substitute the value of d back into Equation 1: \[ 2a + 3\left(-\frac{5}{6}\right) = 3 \Rightarrow 2a - \frac{5}{2} = 3 \Rightarrow 2a = 3 + \frac{5}{2} = \frac{11}{2} \Rightarrow a = \frac{11}{4} \]
Step 4: Calculating S\(_{12}\):
Now we find the sum of the first 12 terms using the formula for S\(_n\) with n=12. \[ S_{12} = \frac{12}{2}[2a + (12-1)d] = 6[2a + 11d] \] Substitute the values of a and d we found: \[ S_{12} = 6\left[2\left(\frac{11}{4}\right) + 11\left(-\frac{5}{6}\right)\right] \] \[ S_{12} = 6\left[\frac{11}{2} - \frac{55}{6}\right] \] To subtract the fractions, find a common denominator (6): \[ S_{12} = 6\left[\frac{33}{6} - \frac{55}{6}\right] = 6\left[-\frac{22}{6}\right] = -22 \]
Step 5: Final Answer:
The sum of the first 12 terms of the AP is -22.