AP & GP SeriesJEE Main

Concept

A progression is a sequence built by a fixed rule. In an AP each term adds a constant difference $d$; in a GP each term multiplies by a constant ratio $r$. Both have neat closed-form sums, and a GP with $|r|<1$ even has a finite infinite sum.

Key formula

Derivation

For an AP, pairing the first and last terms ($a$ and $a+(n-1)d$) gives $n/2$ equal pairs, so $S_n = \tfrac{n}{2}(a_1 + a_n)$.

For a GP, subtracting $rS_n$ from $S_n$ telescopes to $S_n(1-r) = a(1-r^n)$. When $|r|<1$, $r^n \to 0$ and the infinite sum converges to $S_\infty = a/(1-r)$.

Scenarios to explore

• AP & GP Series — Arithmetic & geometric progressions, nth term and sums.

Real-world applications

• Compound interest and loan amortisation (GP).
• Uniformly accelerated motion distances (AP).
• Fractal lengths and repeating-decimal values (infinite GP).

JEE exam tips

• Three numbers in AP: take them as $a-d,\,a,\,a+d$. In GP: $a/r,\,a,\,ar$.
• The arithmetic mean ≥ geometric mean (AM ≥ GM), equal only when all terms match.
• Sum of first $n$ naturals $= n(n+1)/2$ is just an AP with $a=d=1$.

Common mistakes

• Using the AP sum formula for a GP or vice-versa.
• Forgetting the $r=1$ special case in a GP (then $S_n = na$).
• Applying $S_\infty = a/(1-r)$ when $|r| \ge 1$ (it diverges).

Exam traps to avoid

• A GP can alternate in sign when $r<0$ — track the sign of $r^{n-1}$.
• $S_\infty$ exists only for $-1 < r < 1$.