Question

In an arithmetic sequence, the sum of first and third terms is 6 and the sum of second and fourth terms is 20. Then the $11^{\text{th}}$ term is

• A. 67
• B. 62
• C. 57
• D. 73
• E. 66

Hint:

Shortcut Tip: You can bypass finding "$a$" immediately by noticing $(a_2 + a_4) - (a_1 + a_3) = 2d$. Therefore, $20 - 6 = 14 \implies 2d = 14 \implies d=7$.

Answer 1

Answer: (E)

Concept:
The $n$-th term of an Arithmetic Progression (A.P.) is given by the formula $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. We can set up a system of linear equations using the given sums to find $a$ and $d$.

Step 1: Set up the equation for the first sum.
The sum of the first ($a_1$) and third ($a_3$) terms is 6. $$a + (a + 2d) = 6$$ $$2a + 2d = 6$$ Dividing the entire equation by 2 gives: $$a + d = 3$$

Step 2: Set up the equation for the second sum.
The sum of the second ($a_2$) and fourth ($a_4$) terms is 20. $$(a + d) + (a + 3d) = 20$$ $$2a + 4d = 20$$ Dividing the entire equation by 2 gives: $$a + 2d = 10$$

Step 3: Solve for the common difference (d).
Subtract the simplified equation from Step 1 from the equation in
Step 2: $$(a + 2d) - (a + d) = 10 - 3$$ $$d = 7$$

Step 4: Solve for the first term (a).
Substitute the value of $d$ back into the first simplified equation ($a + d = 3$): $$a + 7 = 3$$ $$a = 3 - 7$$ $$a = -4$$

Step 5: Calculate the 11th term.
Now use the standard $n$-th term formula for $a_{11}$: $$a_{11} = a + 10d$$ $$a_{11} = -4 + 10(7)$$ $$a_{11} = -4 + 70 = 66$$ Hence the correct answer is (E) 66.