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Sequences & Series formula sheet

Every formula in the Sequences & Series chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 26 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Arithmetic progression

nth term of an AP
18× asked, last 2026
an=a+(n−1)d,ap−aq=(p−q)dkth from the end=l−(k−1)d\begin{aligned}&a_n=a+(n-1)d,\qquad a_p-a_q=(p-q)d\\&k\text{th from the end}=l-(k-1)d\end{aligned}
a,b,ca,b,c in AP   ⟺  2b=a+c\iff2b=a+c. log⁡a1/kx=klog⁡ax\log_{a^{1/k}}x=k\log_ax.
Sum of n terms
23× asked, last 2026
Sn=n2[2a+(n−1)d]=n2(a+l)S_n=\frac n2\left[2a+(n-1)d\right]=\frac n2(a+l)
Odd nn: Sn=n×S_n=n\times middle term. S30−S10S_{30}-S_{10} has 20 terms. Two APs: am:am′a_m:a_m^\prime = Sn:Sn′S_n:S_n^\prime at n=2m−1n=2m-1.
Term from the sum
5× asked, last 2026
an=Sn−Sn−1 (n≥2),a1=S1Sn=An2+Bn  ⟺  AP with d=2A\begin{aligned}&a_n=S_n-S_{n-1}\ (n\ge2),\qquad a_1=S_1\\&S_n=An^2+Bn\iff\text{AP with }d=2A\end{aligned}
A constant in SnS_n spoils only a1a_1.
AP: symmetric terms
5× asked, last 2026
a−d, a, a+da−3d, a−d, a+d, a+3d  (difference 2d)\begin{aligned}&a-d,\ a,\ a+d\\&a-3d,\ a-d,\ a+d,\ a+3d\ \ (\text{difference }2d)\end{aligned}
The sum fixes aa; product a(a2−d2)a(a^2-d^2). Test each root of dd against all data.
Properties of an AP
8× asked, last 2025
ak+an+1−k=a1+an,Seven−Sodd=n2d (n even)a_k+a_{n+1-k}=a_1+a_n,\qquad S_{\text{even}}-S_{\text{odd}}=\frac n2d\ (n\text{ even})
Adding or scaling keeps an AP; two APs add to an AP.
Arithmetic means
4× asked, last 2026
d=b−an+1,Ak=a+k b−an+1∑Ak=n(a+b)2\begin{aligned}&d=\frac{b-a}{n+1},\qquad A_k=a+k\,\frac{b-a}{n+1}\\&\sum A_k=\frac{n(a+b)}2\end{aligned}
nn means: n+2n+2 terms, n+1n+1 gaps.
Common terms
9× asked, last 2026
d=lcm⁡(d1,d2),last common term≤min⁡(l1,l2)d=\operatorname{lcm}(d_1,d_2),\qquad \text{last common term}\le\min(l_1,l_2)
Start at the first number in both lists. n(A∪B)=n(A)+n(B)−n(A∩B)n(A\cup B)=n(A)+n(B)-n(A\cap B).
Counting and conditional sums
10× asked, last 2026
(n−1)d=l−a∑multiples of k≤N=k m(m+1)2, m=⌊Nk⌋\begin{aligned}&(n-1)d=l-a\\&\sum_{\text{multiples of }k\le N}=k\,\frac{m(m+1)}2,\ m=\left\lfloor\frac Nk\right\rfloor\end{aligned}
TpT_p to TqT_q: q−p+1q-p+1 terms. Alternating jumps: add in pairs.

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Practise Sequences & Series

18 more formulas in 4 topics

Geometric progression, Means and harmonic progression, Arithmetico-geometric and special sums and Telescoping and recursive sequences. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Geometric progression

  • nth term and sum20×
  • GP: symmetric terms6×
  • Products of terms11×
  • Logs of a GP7×
  • Infinite GP15×
  • AP and GP together15×

Means and harmonic progression

  • AM, GM, HM5×
  • AM–GM extremes6×
  • Harmonic progression6×

Arithmetico-geometric and special sums

  • AGP19×
  • Sums of powers18×
  • Alternating sums8×
  • Method of differences10×

Telescoping and recursive sequences

  • Telescoping8×
  • Products in AP17×
  • Quartic denominators8×
  • Root forms2×
  • Linear recurrences16×

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About 245 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

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