Question

Fourth term of an arithmetic progression is $8$. What is the sum of the first $7$ terms of the arithmetic progression?

• A. 7
• B. 64
• C. 56
• D. Cannot be determined

Hint:

If the sum formula has the same linear combination of $a$ and $d$ as a given term, you can directly substitute without finding $a$ or $d$ separately.

Answer 1

The Correct Option is D

Let the first term of the arithmetic progression be $a$ and the common difference be $d$.
The fourth term is: $a + 3d = 8$.
The sum of the first $7$ terms is given by: \[ S_7 = \frac{7}{2} [ 2a + 6d ]. \] We can factor out 2: $S_7 = \frac{7}{2} \times 2 (a + 3d) = 7(a + 3d)$.
But $a + 3d = 8$ (from the given).
Thus $S_7 = 7 \times 8 = 56$. Wait — this gives a unique number, so it \emph{can} be determined.
Therefore, the answer is $\boxed{56}$, matching option (C), not (D). This is a case where the “cannot be determined” option is misleading, as we can indeed compute it.