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Geometric Progression: nth term & sum

Sequences and Series · Class XI

First term a, common ratio r, position n. Try |r| < 1 vs |r| > 1 for growth or decay.

2
type a value
-1010
1.5
type a value
-22
5
type a value
125
Live values
  • n-th term10.125
  • Sum to n terms (r ≠ 1)26.375
  • Infinite sum (|r| < 1):
xy
  • a · r^(x − 1)

Formulas in this lab

  • n-th term
    an=a⋅rn−1a_n = a \cdot r^{n-1}
  • Sum to n terms (r ≠ 1)
    Sn=arn−1r−1S_n = a\dfrac{r^n - 1}{r - 1}
  • Infinite sum (|r| < 1)
    S∞=a1−rS_\infty = \dfrac{a}{1 - r}
Tip: |r| < 1 makes the curve flatten , infinite sum converges only in that range.

Frequently asked questions

▶What is a geometric progression?

A GP is a sequence where each term is the previous one multiplied by a fixed ratio $r$. The $n$th term is $a_n = ar^{n-1}$ and the sum is $S_n = a\frac{r^n - 1}{r - 1}$ when $r\ne 1$. For $a = 3$, $r = 2$, the 5th term is $48$ and $S_5 = 93$.

▶How do I use the geometric progression lab?

Slide first term $a$, ratio $r$ and length $n$. The lab returns $a_n$ and $S_n$ and lists terms so you can match the formula to the pattern. Try $a = 1$, $r = 0.5$, $n = 10$ to see the infinite sum approach $2$ — a preview of $S_\infty = a/(1-r)$.

▶When does the infinite sum $S_\infty = a/(1-r)$ converge?

Only when $|r| < 1$, i.e. the terms shrink. For $|r| \ge 1$ the sum blows up. Forgetting this condition is the most common GP error on JEE and board papers. The lab lets you nudge $r$ past $1$ to see the sum explode.

▶Where do GPs power real-world growth?

Compound interest at $7\%$ p.a. turns Rs $1000$ into $1000\times 1.07^n$ after $n$ years — a GP. Population growth, viral content sharing on WhatsApp, and bacterial cultures all follow GP-style multiplication. Slide $r$ near $1.1$ and feel the exponential tail.