Sequences and Series
Board: JEE Main |Exam:JEEMain
JEEMainstudynotesforSequencesandSeriesbyAjayYadav.
KeyConcepts
AP,GP,HP
AP:a,a+d,a+2d,…Sum=n/2[2a+(n-1)d].GP:a,ar,ar²,…Sum=a(rⁿ-1)/(r-1).HP:1/a,1/(a+d),…
AM,GM,HM
AM=(a+b)/2,GM=√(ab),HM=2ab/(a+b).AM≥GM≥HM.Equalityiffa=b.
SigmaNotation
Σn=n(n+1)/2.Σn²=n(n+1)(2n+1)/6.Σn³=[n(n+1)/2]².
Arithmetico-GeometricSeries
a+(a+d)r+(a+2d)r²+…Sumformulausingsubtract-and-multiplymethod.
ImportantFormulas
| APsum | S₅=n/2[2a+(n-1)d] |
| GPinfinite | S∞=a/(1-r),|r|<1 |
| AM ≥ GM | (a+b)/2 ≥ √ab |
| Σn² | n(n+1)(2n+1)/6 |
Solved Examples
Example 1: Sum to n: 1×2+2×3+3×4+…
Solution: Tₔ=r(r+1)=r²+r. S₅=Σr²+Σr=n(n+1)(2n+1)/6+n(n+1)/2 = n(n+1)(n+2)/3.
Practice Questions
- Sum: 1³+2³+…+n³.
- Insert 4 GMs between 2 and 64.
- If AM=10, GM=6, find HM.
- Find sum of infinite GP: 1, -1/2, 1/4, -1/8,…
- Sum: 1/1×2 + 1/2×3 + 1/3×4 + … + 1/n(n+1).