Question

Let $S_n$ denote the sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \dots$. If $S_{n+3} - S_n = 13n + 7$ for all $n$, what is the value of $a_{13} - a_{10}$?

• A. 13
• B. 137
• C. 46
• D. 12

Hint:

To find the difference between terms separated by a gap of $d$, evaluate the sum-term relation at $n$ and $n-1$.
The intermediate terms will cancel out, leaving the direct difference of interest:
\[ (S_{n+3} - S_n) - (S_{n+2} - S_{n-1}) = a_{n+3} - a_n \]

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The difference between two sums $S_m - S_n$ (where $m > n$) gives the sum of the terms from $a_{n+1}$ to $a_m$. Specifically, $S_{n+3} - S_n = a_{n+3} + a_{n+2} + a_{n+1}$.
Step 2: Setting up the Equations:
We are given $a_{n+3} + a_{n+2} + a_{n+1} = 13n + 7$.
To find $a_{13} - a_{10}$, let's evaluate the expression for two consecutive values of $n$:
For $n = 10$: $a_{13} + a_{12} + a_{11} = 13(10) + 7 = 137$.
For $n = 9$: $a_{12} + a_{11} + a_{10} = 13(9) + 7 = 117 + 7 = 124$.

Step 3: Calculating the Difference:
Subtract the second equation from the first: $(a_{13} + a_{12} + a_{11}) - (a_{12} + a_{11} + a_{10}) = 137 - 124$ $a_{13} - a_{10} = 13$.
Step 4: Final Answer:
The value of $a_{13} - a_{10}$ is 13.