Finding the First Term of an AP from a Sum

This runs the AP sum formula in reverse: with the sum, the number of terms, and the common difference known, the formula becomes a linear equation in the first term $a$.

The sum of the first $10$ terms of an arithmetic progression is $210$. Given that the common difference is $4$, find the first term. [3]

The working

Step 1, substitute into $S_n = \dfrac{n}{2}\big[2a + (n - 1)d\big]$ with $S_{10} = 210$, $n = 10$, $d = 4$: $210 = \frac{10}{2}\big[2a + 9(4)\big] \quad \text{(M1)}$

Step 2, simplify: $210 = 5\big[2a + 36\big] \;\Rightarrow\; 210 = 10a + 180 \quad \text{(M1)}$

Step 3, solve for $a$: $10a = 30 \;\Rightarrow\; a = 3 \quad \text{(A1)}$

So the first term is $3$.

Where the marks are won and lost

• Use $(n - 1)d = 9 \times 4 = 36$, not $10 \times 4$. The off-by-one on $(n-1)$ is the classic error.
• Expand carefully: $5[2a + 36] = 10a + 180$. Then it’s a straightforward linear equation.
• Only $a$ is unknown, everything else is given, so treat it as solving for one variable.

Common mistakes

• Using $10 \times 4$ instead of $9 \times 4$ inside the bracket.
• Arithmetic slips expanding $5[2a + 36]$.
• Confusing the sum $S_{10}$ with the 10th term.

Full method: Arithmetic Progressions notes. Topic home: Series pillar.