IIT JEE Main Maths -Unit 6- Sum of n Terms of Special Series- Study Notes-New Syllabus

Sigma (Σ) Notation and Summation of Series (A.P., G.P., and A.G.P.)

The symbol \( \Sigma \) (capital sigma) represents the sum of a sequence of terms.

$ \sum_{r=1}^{n} a_r = a_1 + a_2 + a_3 + \dots + a_n $

Here, \( a_r \) is the general term, and the index \( r \) varies from 1 to \( n \).

Basic Sigma Formulas (Very Important)

General Rule: $ \sum_{r=1}^{n} (a + br + cr^2 + \dots) = a\sum 1 + b\sum r + c\sum r^2 + \dots $

 Summation of an Arithmetic Series (A.P.)

If the series is in A.P.: $ a,\, a + d,\, a + 2d,\, \dots,\, a + (n – 1)d $ then,

$ \sum_{r=1}^{n} (a + (r – 1)d) = \dfrac{n}{2}[2a + (n – 1)d] $

Summation of a Geometric Series (G.P.)

If the series is in G.P.: $ a,\, ar,\, ar^2,\, ar^3,\, \dots $ then,

$ \sum_{r=1}^{n} ar^{r – 1} = \dfrac{a(r^n – 1)}{r – 1}, \quad (r \ne 1) $

and if \( |r| < 1 \): $ S_\infty = \dfrac{a}{1 – r} $

 Summation of an Arithmetic–Geometric Series (A.G.P.)

Series of the form: $ a + (a + d)r + (a + 2d)r^2 + \dots $ have the sum:

$ S_n = \dfrac{a(1 – r^n)}{1 – r} + \dfrac{dr(1 – nr^{n – 1} + (n – 1)r^n)}{(1 – r)^2} $

and if \( |r| < 1 \):

$ S_\infty = \dfrac{a}{1 – r} + \dfrac{dr}{(1 – r)^2} $

 Properties of Sigma Notation

• \( \displaystyle \sum_{r=1}^{n} (a_r + b_r) = \sum a_r + \sum b_r \)
• \( \displaystyle \sum_{r=1}^{n} ka_r = k\sum a_r \)
• \( \displaystyle \sum_{r=1}^{n} (r + 1) = \sum r + \sum 1 = \dfrac{n(n + 1)}{2} + n \)
• \( \displaystyle \sum_{r=1}^{n} (n – r + 1) = \sum_{r=1}^{n} r = \dfrac{n(n + 1)}{2} \)

Shortcut Sigma Expansions (for Quick JEE Solving)

• \( \displaystyle \sum_{r=1}^{n} (a + br) = na + b\dfrac{n(n + 1)}{2} \)
• \( \displaystyle \sum_{r=1}^{n} (ar + br^2) = a\dfrac{n(n + 1)}{2} + b\dfrac{n(n + 1)(2n + 1)}{6} \)
• \( \displaystyle \sum_{r=1}^{n} r(r + 1) = \dfrac{n(n + 1)(n + 2)}{3} \)
• \( \displaystyle \sum_{r=1}^{n} r(r – 1) = \dfrac{n(n + 1)(n – 1)}{3} \)

Key Takeaways for JEE:

• Break complex sums into known forms using basic Σ formulas.
• Use Σ identities to simplify polynomial and mixed series.
• Always check for G.P. or A.G.P. patterns before expanding manually.
• For infinite geometric or A.G.P. series, ensure \( |r| < 1 \) for convergence.
• Remember the power sum formulas — they appear often in JEE algebra and calculus limits.

 Common JEE Mistakes to Avoid

• Forgetting to apply limits correctly when starting from \( r = 0 \) or another index.
• Using A.P. formulas for geometric or mixed progressions.
• For infinite sums, forgetting to verify \( |r| < 1 \).