JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
SEQUENCE AND SERIES
— JEE MATH APEX —
JEE MATH APEX
1. Arithmetic Progression (AP)
📌 General Term & Sum of AP
• General form: a, a+d, a+2d, a+3d, …
• nth term: an = a + (n−1)d
• Last term: l = a + (n−1)d
• Sum of n terms: Sn = n/2 [2a + (n−1)d]
• Sum using first & last term: Sn = n/2 (a + l)
• nth term from end: l − (n−1)d
• nth term: an = a + (n−1)d
• Last term: l = a + (n−1)d
• Sum of n terms: Sn = n/2 [2a + (n−1)d]
• Sum using first & last term: Sn = n/2 (a + l)
• nth term from end: l − (n−1)d
JEE MATH APEX
2. Properties of AP
📌 Important Properties
• If a, b, c are in AP: 2b = a + c
• Sum of terms equidistant from ends is constant: a₁ + an = a₂ + an−1 = …
• If a constant is added/subtracted to each term: AP remains AP
• If each term is multiplied/divided by constant: AP remains AP
• Sum of first n odd numbers: 1 + 3 + 5 + … + (2n−1) = n²
• Sum of first n even numbers: 2 + 4 + 6 + … + 2n = n(n+1)
• Sum of terms equidistant from ends is constant: a₁ + an = a₂ + an−1 = …
• If a constant is added/subtracted to each term: AP remains AP
• If each term is multiplied/divided by constant: AP remains AP
• Sum of first n odd numbers: 1 + 3 + 5 + … + (2n−1) = n²
• Sum of first n even numbers: 2 + 4 + 6 + … + 2n = n(n+1)
JEE MATH APEX
3. Geometric Progression (GP)
📌 General Term & Sum of GP
• General form: a, ar, ar², ar³, …
• nth term: an = arn−1
• Sum of n terms (r ≠ 1): Sn = a(1 − rn)/(1 − r)
• Sum of n terms (r ≠ 1): Sn = a(rn − 1)/(r − 1) (if r > 1)
• If r = 1: Sn = na
• nth term from end: arn−1 · (1/r)k−1
• nth term: an = arn−1
• Sum of n terms (r ≠ 1): Sn = a(1 − rn)/(1 − r)
• Sum of n terms (r ≠ 1): Sn = a(rn − 1)/(r − 1) (if r > 1)
• If r = 1: Sn = na
• nth term from end: arn−1 · (1/r)k−1
JEE MATH APEX
4. Properties of GP
📌 Important Properties
• If a, b, c are in GP: b² = ac
• Product of terms equidistant from ends is constant: a₁·an = a₂·an−1 = …
• If each term is multiplied/divided by constant: GP remains GP
• If each term is raised to same power: GP remains GP
• Product of first n terms: Pn = an · rn(n−1)/2
• Product of terms equidistant from ends is constant: a₁·an = a₂·an−1 = …
• If each term is multiplied/divided by constant: GP remains GP
• If each term is raised to same power: GP remains GP
• Product of first n terms: Pn = an · rn(n−1)/2
JEE MATH APEX
5. Sum of Infinite GP
📌 Infinite Geometric Series
• Sum to infinity (|r| < 1): S∞ = a / (1 − r)
• Condition: |r| < 1 (otherwise series diverges)
• Sum of infinite GP with first term a and common ratio r: a + ar + ar² + … = a/(1−r)
• Example: 1 + 1/2 + 1/4 + 1/8 + … = 2
• Infinite GP with negative ratio: 1 − 1/2 + 1/4 − 1/8 + … = 2/3
• Condition: |r| < 1 (otherwise series diverges)
• Sum of infinite GP with first term a and common ratio r: a + ar + ar² + … = a/(1−r)
• Example: 1 + 1/2 + 1/4 + 1/8 + … = 2
• Infinite GP with negative ratio: 1 − 1/2 + 1/4 − 1/8 + … = 2/3
JEE MATH APEX
6. Harmonic Progression (HP)
📌 Harmonic Progression
• If reciprocals of terms are in AP: 1/a, 1/(a+d), 1/(a+2d), …
• nth term of HP: an = 1/[a + (n−1)d]
• If a, b, c are in HP: b = 2ac/(a + c)
• Harmonic Mean (HM) of a and b: H = 2ab/(a + b)
• Relation: AM ≥ GM ≥ HM
• nth term of HP: an = 1/[a + (n−1)d]
• If a, b, c are in HP: b = 2ac/(a + c)
• Harmonic Mean (HM) of a and b: H = 2ab/(a + b)
• Relation: AM ≥ GM ≥ HM
JEE MATH APEX
7. Arithmetic Mean (AM)
📌 AM & Insertion of AMs
• AM of a and b: AM = (a + b) / 2
• Inserting n AMs between a and b:
Common difference: d = (b − a) / (n + 1)
• n AMs: A₁ = a + d, A₂ = a + 2d, …, An = a + nd
• Sum of n AMs: n(a + b)/2
• If A₁, A₂, …, An are n AMs: A₁ + A₂ + … + An = n(a+b)/2
• Inserting n AMs between a and b:
Common difference: d = (b − a) / (n + 1)
• n AMs: A₁ = a + d, A₂ = a + 2d, …, An = a + nd
• Sum of n AMs: n(a + b)/2
• If A₁, A₂, …, An are n AMs: A₁ + A₂ + … + An = n(a+b)/2
JEE MATH APEX
8. Geometric Mean (GM)
📌 GM & Insertion of GMs
• GM of a and b: GM = √(ab)
• Inserting n GMs between a and b:
Common ratio: r = (b/a)1/(n+1)
• n GMs: G₁ = ar, G₂ = ar², …, Gn = arn
• Product of n GMs: (ab)n/2
• If G₁, G₂, …, Gn are n GMs: G₁ × G₂ × … × Gn = (ab)n/2
• Inserting n GMs between a and b:
Common ratio: r = (b/a)1/(n+1)
• n GMs: G₁ = ar, G₂ = ar², …, Gn = arn
• Product of n GMs: (ab)n/2
• If G₁, G₂, …, Gn are n GMs: G₁ × G₂ × … × Gn = (ab)n/2
JEE MATH APEX
9. AM-GM-HM Inequality
📌 Important Inequalities
• For two positive numbers: AM ≥ GM ≥ HM
• (a + b)/2 ≥ √(ab) (AM ≥ GM)
• √(ab) ≥ 2ab/(a + b) (GM ≥ HM)
• Equality holds when a = b
• For n positive numbers: (a₁ + a₂ + … + an)/n ≥ (a₁·a₂·…·an)1/n
• Weighted AM ≥ Weighted GM
• (a + b)/2 ≥ √(ab) (AM ≥ GM)
• √(ab) ≥ 2ab/(a + b) (GM ≥ HM)
• Equality holds when a = b
• For n positive numbers: (a₁ + a₂ + … + an)/n ≥ (a₁·a₂·…·an)1/n
• Weighted AM ≥ Weighted GM
JEE MATH APEX
10. Special Series
📌 Sum of Natural Numbers, Squares, Cubes
• Sum of first n natural numbers: Σn = n(n+1)/2
• Sum of squares: Σn² = n(n+1)(2n+1)/6
• Sum of cubes: Σn³ = [n(n+1)/2]²
• Σn⁴ = n(n+1)(2n+1)(3n²+3n−1)/30
• Sum of first n odd natural numbers: n²
• Sum of first n even natural numbers: n(n+1)
• Sum of squares: Σn² = n(n+1)(2n+1)/6
• Sum of cubes: Σn³ = [n(n+1)/2]²
• Σn⁴ = n(n+1)(2n+1)(3n²+3n−1)/30
• Sum of first n odd natural numbers: n²
• Sum of first n even natural numbers: n(n+1)
JEE MATH APEX
11. Arithmetic-Geometric Progression (AGP)
📌 Sum of AGP
• General form: a, (a+d)r, (a+2d)r², (a+3d)r³, …
• nth term: Tn = [a + (n−1)d] · rn−1
• Sum of n terms: Sn = a/(1−r) + dr(1 − rn−1)/(1−r)² − [a + (n−1)d]rn/(1−r)
• Sum to infinity (|r| < 1): S∞ = a/(1−r) + dr/(1−r)²
• Example: 1 + 2x + 3x² + 4x³ + … = 1/(1−x)² (|x| < 1)
• nth term: Tn = [a + (n−1)d] · rn−1
• Sum of n terms: Sn = a/(1−r) + dr(1 − rn−1)/(1−r)² − [a + (n−1)d]rn/(1−r)
• Sum to infinity (|r| < 1): S∞ = a/(1−r) + dr/(1−r)²
• Example: 1 + 2x + 3x² + 4x³ + … = 1/(1−x)² (|x| < 1)
JEE MATH APEX
12. Sum of Special Products
📌 Products & Advanced Sums
• Σn(n+1) = n(n+1)(n+2)/3
• Σn(n+1)(n+2) = n(n+1)(n+2)(n+3)/4
• Σ1/[n(n+1)] = n/(n+1)
• Σ(2n−1)² = n(2n−1)(2n+1)/3
• Sum of infinite series: 1/(1·2) + 1/(2·3) + 1/(3·4) + … = 1
• Σn(n+1)(n+2) = n(n+1)(n+2)(n+3)/4
• Σ1/[n(n+1)] = n/(n+1)
• Σ(2n−1)² = n(2n−1)(2n+1)/3
• Sum of infinite series: 1/(1·2) + 1/(2·3) + 1/(3·4) + … = 1
JEE MATH APEX
13. Relation Between AM, GM, HM
📌 AM-GM-HM Relationship
• For two positive numbers a and b:
AM = (a + b)/2
GM = √(ab)
HM = 2ab/(a + b)
• GM² = AM × HM
• AM − GM = (√a − √b)²/2 ≥ 0
• (AM − HM) = [(a − b)²]/[2(a + b)]
• If AM = GM = HM, then a = b
AM = (a + b)/2
GM = √(ab)
HM = 2ab/(a + b)
• GM² = AM × HM
• AM − GM = (√a − √b)²/2 ≥ 0
• (AM − HM) = [(a − b)²]/[2(a + b)]
• If AM = GM = HM, then a = b
JEE MATH APEX
14. Method of Differences
📌 Sum Using Difference Method
• For series where difference of consecutive terms follows a pattern:
Sn = Σ Tk = Σ [f(k) − f(k−1)]
• Sn = f(n) − f(0)
• Used for series like: 1·2 + 2·3 + 3·4 + … + n(n+1)
• Tn = n(n+1) = n² + n → Sum = Σn² + Σn
• Vn method: Tn = an − an−1
Sn = Σ Tk = Σ [f(k) − f(k−1)]
• Sn = f(n) − f(0)
• Used for series like: 1·2 + 2·3 + 3·4 + … + n(n+1)
• Tn = n(n+1) = n² + n → Sum = Σn² + Σn
• Vn method: Tn = an − an−1
✅ SEQUENCE AND SERIES ✅
— JEE MATH APEX —
— JEE MATH APEX —
JEE MATH APEX 
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