Consider an A.P with $n$ terms such that the first term is $a$, the last term is $l$ and the common difference $d$. Then the sum of all $n$ terms is given by

$$S_n = a + (a + d) + (a + 2d) +\cdots + (l\mathbin{-}2d) + (l\mathbin{-}d) + l$$

Writing the sum in reverse order, we get

$$S_n = l + (l\mathbin{-}d) + (l\mathbin{-}2d) + \cdots + (a + 2d) + (a + d) + a$$

Now adding corresponding terms in the two formulas above gives

$$2S_n = (a + l) + (a + l) + (a + l) + \cdots + (a + l) + (a + l) + (a + l)$$

and since there are $n$ terms all being $(a + l)$ we get

$$2S_n = n(a + l)$$

and

$$S_n = n \frac{(a + l)}{2}$$

Thus the sum of terms in an A.P is the product of number of terms by the average of the first and last terms.

If we substitute $l = a + (n\mathbin{-}1)d$ in the final formula for $S_n$ we get:

$$S_n = n \frac{a + a + (n\mathbin{-}1)d}{2} = n\frac{2a + nd\mathbin{-} d}{2} = \frac{d}{2}n^2 + \frac{2a\mathbin{-} d}{2}n$$

The last formula is quadratic in $n$ thus plotting various values of $n$ against corresponding values of $S_n$ for any arithmetic series gives a parabolic curve.

Exercises:

1 For a given series the sum of the first $n$ terms is $pn^2 + qn$ where $p, q$ are constants. Prove that the series in an A.P. and find its first term and common difference in terms of $p$ and $q$.

2. Without using any of the above formulas for sum of terms in an A.P. find the sum of all:

a) positive integers less than 101

b) odd positive integers less than 101

c) even positive integers less than 101

Hint: in each case form pairs of terms: first and last, second and penultimate etc.

3. Using the fact that sum of an A.P. is equal to the number of terms times the average of first and last terms we quickly see that $$1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}$$ Prove this formula geometrically. Hint: represent 1 as a unit square, two as two unit squares stacked one on the other etc.

4 Read up on triangular numbers and then find the tenth triangular number.

5. Two men both started working different jobs at the beginning of 2025. Their salaries were \$17,000 and \$15,000 per year with annual increments of \$500 and \$1,000 respectively. In what year will they earn the same salary? How many years must pass before they would have earned the same cumulative amount?