Arithmetic Progression: nth term & sum

An AP is a sequence where consecutive terms differ by a fixed amount $d$, called the common difference. The $n$th term is $a_n = a + (n-1)d$, and the sum of the first $n$ terms is $S_n = \frac{n}{2}(2a + (n-1)d)$. So for $a = 2$, $d = 3$, the 10th term is $29$.

Slide the first term $a$, common difference $d$ and number of terms $n$. The lab returns $a_n$, $S_n$ and lists the first few terms so you can verify by hand. Try $a = 5$, $d = 2$, $n = 20$: $a_{20} = 43$ and $S_{20} = 480$.

Auto-rickshaw fares with a flag-down rate of Rs $30$ plus Rs $15$ per km form an AP. EMIs reducing by a fixed amount each month, seat numbers in a movie hall row, and stadium ticket prices for successive blocks are all APs. Sums help you budget a long-haul journey.

AP has a constant difference: $5, 8, 11, 14$ (add 3). GP has a constant ratio: $5, 10, 20, 40$ (multiply by 2). Mixing them up is the most common JEE sequences-and-series mistake. The companion GP lab makes the contrast vivid: AP grows linearly, GP grows exponentially.