Cheat Sheet: Sequences and Series | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Sequences and Series Cheat Sheet 
(EduRev) 
Sequences 
Key De?nitions 
? Sequence: Ordered list of numbers (minimum 3 terms), e.g., {a
1
, a
2
, a
3
, ...}. 
? Finite Sequence: Fixed number of terms, e.g., {1, 3, 5, 7}. 
? In?nite Sequence: Continues inde?nitely, e.g., {1, 2, 3, ...}. 
? Rule: Formula for n-th term, e.g., T
n
 = 2n + 1 for {3, 5, 7, ...}. 
? Notation: T
n
 denotes n-th term, e.g., T
5
 for 5th term. 
Series 
Key De?nitions 
? Series: Sum of sequence terms, S
n
 = T
1
 + T
2
 + ... + T
n
. 
? Sigma Notation (S): Summation, e.g., S
n=1
k
 n = 1 + 2 + ... + k. 
? Pi Notation (?): Product, e.g., ?
n=1
k
 n = 1 × 2 × ... × k = k!. 
Sigma Properties 
? S
i=1
k
 a = ka (a is constant). 
? S
i=1
k
 (a
i
 ± b
i
) = S
i=1
k
 a
i
 ± S
i=1
k
 b
i
. 
Page 2


Sequences and Series Cheat Sheet 
(EduRev) 
Sequences 
Key De?nitions 
? Sequence: Ordered list of numbers (minimum 3 terms), e.g., {a
1
, a
2
, a
3
, ...}. 
? Finite Sequence: Fixed number of terms, e.g., {1, 3, 5, 7}. 
? In?nite Sequence: Continues inde?nitely, e.g., {1, 2, 3, ...}. 
? Rule: Formula for n-th term, e.g., T
n
 = 2n + 1 for {3, 5, 7, ...}. 
? Notation: T
n
 denotes n-th term, e.g., T
5
 for 5th term. 
Series 
Key De?nitions 
? Series: Sum of sequence terms, S
n
 = T
1
 + T
2
 + ... + T
n
. 
? Sigma Notation (S): Summation, e.g., S
n=1
k
 n = 1 + 2 + ... + k. 
? Pi Notation (?): Product, e.g., ?
n=1
k
 n = 1 × 2 × ... × k = k!. 
Sigma Properties 
? S
i=1
k
 a = ka (a is constant). 
? S
i=1
k
 (a
i
 ± b
i
) = S
i=1
k
 a
i
 ± S
i=1
k
 b
i
. 
Arithmetic Progression (AP) 
Formulas 
? n-th Term: T
n
 = a + (n-1)d, a = ?rst term, d = common difference. 
? n-th Term from End: T
m-n+1
 = T
m
 - (n-1)d, m = total terms. 
? Sum of n Terms: S
n
 = n/2 [2a + (n-1)d] = n/2 (a + T
n
). 
? Arithmetic Mean: AM = (a + b)/2 (two terms); AM = (a
1
 + ... + a
n
)/n (n terms). 
? n AMs between a and b: d = (b - a)/(n+1), terms = a + d, a + 2d, ..., a + nd. 
? Properties: 
? If T
m
 = n and T
n
 = m, then T
m+n
 = 0. 
? Sum of equidistant terms: T
k
 + T
n-k+1
 = a + T
n
. 
? Multiplying/dividing terms by constant C gives AP with difference Cd or 
d/C. 
? S
n
 = An
2
 + Bn, common difference = 2A. 
Geometric Progression (GP) 
Formulas 
? n-th Term: T
n
 = ar
n-1
, a = ?rst term, r = common ratio. 
? Property: T
n
 = v(T
n-1
 T
n+1
). 
? Sum of n Terms: S
n
 = a(1 - r
n
)/(1 - r) (r ? 1); S
n
 = na (r = 1). 
? In?nite GP Sum: S
8
 = a/(1 - r), for |r| < 1. 
? Geometric Mean: GM = v(ab) (two terms); GM = (a
1
 a
2
 ... a
n
)
1/n
 (n terms). 
? n GMs between a and b: r = (b/a)
1/(n+1)
, terms = ar, ar
2
, ..., ar
n
. 
Page 3


Sequences and Series Cheat Sheet 
(EduRev) 
Sequences 
Key De?nitions 
? Sequence: Ordered list of numbers (minimum 3 terms), e.g., {a
1
, a
2
, a
3
, ...}. 
? Finite Sequence: Fixed number of terms, e.g., {1, 3, 5, 7}. 
? In?nite Sequence: Continues inde?nitely, e.g., {1, 2, 3, ...}. 
? Rule: Formula for n-th term, e.g., T
n
 = 2n + 1 for {3, 5, 7, ...}. 
? Notation: T
n
 denotes n-th term, e.g., T
5
 for 5th term. 
Series 
Key De?nitions 
? Series: Sum of sequence terms, S
n
 = T
1
 + T
2
 + ... + T
n
. 
? Sigma Notation (S): Summation, e.g., S
n=1
k
 n = 1 + 2 + ... + k. 
? Pi Notation (?): Product, e.g., ?
n=1
k
 n = 1 × 2 × ... × k = k!. 
Sigma Properties 
? S
i=1
k
 a = ka (a is constant). 
? S
i=1
k
 (a
i
 ± b
i
) = S
i=1
k
 a
i
 ± S
i=1
k
 b
i
. 
Arithmetic Progression (AP) 
Formulas 
? n-th Term: T
n
 = a + (n-1)d, a = ?rst term, d = common difference. 
? n-th Term from End: T
m-n+1
 = T
m
 - (n-1)d, m = total terms. 
? Sum of n Terms: S
n
 = n/2 [2a + (n-1)d] = n/2 (a + T
n
). 
? Arithmetic Mean: AM = (a + b)/2 (two terms); AM = (a
1
 + ... + a
n
)/n (n terms). 
? n AMs between a and b: d = (b - a)/(n+1), terms = a + d, a + 2d, ..., a + nd. 
? Properties: 
? If T
m
 = n and T
n
 = m, then T
m+n
 = 0. 
? Sum of equidistant terms: T
k
 + T
n-k+1
 = a + T
n
. 
? Multiplying/dividing terms by constant C gives AP with difference Cd or 
d/C. 
? S
n
 = An
2
 + Bn, common difference = 2A. 
Geometric Progression (GP) 
Formulas 
? n-th Term: T
n
 = ar
n-1
, a = ?rst term, r = common ratio. 
? Property: T
n
 = v(T
n-1
 T
n+1
). 
? Sum of n Terms: S
n
 = a(1 - r
n
)/(1 - r) (r ? 1); S
n
 = na (r = 1). 
? In?nite GP Sum: S
8
 = a/(1 - r), for |r| < 1. 
? Geometric Mean: GM = v(ab) (two terms); GM = (a
1
 a
2
 ... a
n
)
1/n
 (n terms). 
? n GMs between a and b: r = (b/a)
1/(n+1)
, terms = ar, ar
2
, ..., ar
n
. 
Harmonic Progression (HP) 
Formulas 
? n-th Term: 1/a
n
 = 1/a + (n-1)d, where 1/a
n
 forms AP . 
? Harmonic Mean: HM = 2ab/(a + b) (two terms); HM = n/(1/a
1
 + ... + 1/a
n
) (n terms). 
? n HMs between a and b: 1/H
i
 = 1/a + i(a - b)/[(n+1)ab], i = 1, 2, ..., n. 
? Notes: No general sum formula; 0 cannot be a term. 
? Relation: AM = GM = HM, equality when numbers are equal. 
Arithmetic-Geometric Progression (AGP) 
Formulas 
? Form: ab, (a+d)br, (a+2d)br
2
, ... 
? Sum of n Terms: S
n
 = ab/(1-r) + dbr(1 - r
n-1
)/(1-r)
2
 - [a + (n-1)d]br
n
/(1-r) (r ? 1). 
? In?nite Sum: S
8
 = ab/(1-r) + dbr/(1-r)
2
, for |r| < 1. 
Special Series 
Formulas 
? Sum of n natural numbers: Sn = n(n+1)/2. 
? Sum of n odd numbers: S(2r-1) = n
2
. 
? Sum of n even numbers: S(2r) = n(n+1). 
? Sum of squares: Sn
2
 = n(n+1)(2n+1)/6. 
? Sum of cubes: Sn
3
 = [n(n+1)/2]
2
. 
? Special Sums: 
? S[1/(n(n+1))] = n/(n+1). 
Page 4


Sequences and Series Cheat Sheet 
(EduRev) 
Sequences 
Key De?nitions 
? Sequence: Ordered list of numbers (minimum 3 terms), e.g., {a
1
, a
2
, a
3
, ...}. 
? Finite Sequence: Fixed number of terms, e.g., {1, 3, 5, 7}. 
? In?nite Sequence: Continues inde?nitely, e.g., {1, 2, 3, ...}. 
? Rule: Formula for n-th term, e.g., T
n
 = 2n + 1 for {3, 5, 7, ...}. 
? Notation: T
n
 denotes n-th term, e.g., T
5
 for 5th term. 
Series 
Key De?nitions 
? Series: Sum of sequence terms, S
n
 = T
1
 + T
2
 + ... + T
n
. 
? Sigma Notation (S): Summation, e.g., S
n=1
k
 n = 1 + 2 + ... + k. 
? Pi Notation (?): Product, e.g., ?
n=1
k
 n = 1 × 2 × ... × k = k!. 
Sigma Properties 
? S
i=1
k
 a = ka (a is constant). 
? S
i=1
k
 (a
i
 ± b
i
) = S
i=1
k
 a
i
 ± S
i=1
k
 b
i
. 
Arithmetic Progression (AP) 
Formulas 
? n-th Term: T
n
 = a + (n-1)d, a = ?rst term, d = common difference. 
? n-th Term from End: T
m-n+1
 = T
m
 - (n-1)d, m = total terms. 
? Sum of n Terms: S
n
 = n/2 [2a + (n-1)d] = n/2 (a + T
n
). 
? Arithmetic Mean: AM = (a + b)/2 (two terms); AM = (a
1
 + ... + a
n
)/n (n terms). 
? n AMs between a and b: d = (b - a)/(n+1), terms = a + d, a + 2d, ..., a + nd. 
? Properties: 
? If T
m
 = n and T
n
 = m, then T
m+n
 = 0. 
? Sum of equidistant terms: T
k
 + T
n-k+1
 = a + T
n
. 
? Multiplying/dividing terms by constant C gives AP with difference Cd or 
d/C. 
? S
n
 = An
2
 + Bn, common difference = 2A. 
Geometric Progression (GP) 
Formulas 
? n-th Term: T
n
 = ar
n-1
, a = ?rst term, r = common ratio. 
? Property: T
n
 = v(T
n-1
 T
n+1
). 
? Sum of n Terms: S
n
 = a(1 - r
n
)/(1 - r) (r ? 1); S
n
 = na (r = 1). 
? In?nite GP Sum: S
8
 = a/(1 - r), for |r| < 1. 
? Geometric Mean: GM = v(ab) (two terms); GM = (a
1
 a
2
 ... a
n
)
1/n
 (n terms). 
? n GMs between a and b: r = (b/a)
1/(n+1)
, terms = ar, ar
2
, ..., ar
n
. 
Harmonic Progression (HP) 
Formulas 
? n-th Term: 1/a
n
 = 1/a + (n-1)d, where 1/a
n
 forms AP . 
? Harmonic Mean: HM = 2ab/(a + b) (two terms); HM = n/(1/a
1
 + ... + 1/a
n
) (n terms). 
? n HMs between a and b: 1/H
i
 = 1/a + i(a - b)/[(n+1)ab], i = 1, 2, ..., n. 
? Notes: No general sum formula; 0 cannot be a term. 
? Relation: AM = GM = HM, equality when numbers are equal. 
Arithmetic-Geometric Progression (AGP) 
Formulas 
? Form: ab, (a+d)br, (a+2d)br
2
, ... 
? Sum of n Terms: S
n
 = ab/(1-r) + dbr(1 - r
n-1
)/(1-r)
2
 - [a + (n-1)d]br
n
/(1-r) (r ? 1). 
? In?nite Sum: S
8
 = ab/(1-r) + dbr/(1-r)
2
, for |r| < 1. 
Special Series 
Formulas 
? Sum of n natural numbers: Sn = n(n+1)/2. 
? Sum of n odd numbers: S(2r-1) = n
2
. 
? Sum of n even numbers: S(2r) = n(n+1). 
? Sum of squares: Sn
2
 = n(n+1)(2n+1)/6. 
? Sum of cubes: Sn
3
 = [n(n+1)/2]
2
. 
? Special Sums: 
? S[1/(n(n+1))] = n/(n+1). 
? S[1/(n(n+1)(n+2))] = 1/4 - 1/[2(n+1)(n+2)]. 
? S[1/(n(n+1)(n+2)(n+3))] = 1/18 - 1/[3(n+1)(n+2)(n+3)], S
8
 = 1/18. 
AM, GM, HM Relations 
Formulas 
? Inequality: AM = GM = HM, equality when numbers are equal. 
? Geometric Relation: G
2
 = AH. 
? Quadratic Equation: For two numbers a, b: x
2
 - 2Ax + G
2
 = 0. 
? Cubic Equation: For three numbers a, b, c: x
3
 - 3Ax
2
 + (3G
3
/H)x - G
3
 = 0. 
? AM of m-th Power: (a
1
m
 + ... + a
n
m
)/n > (
m
v(a
1
 + ... + a
n
))
m
 if m ? [0,1]; reverse if m ? (0,1). 
Problem-Solving Tactics 
? Write out terms to spot patterns without simplifying. 
? AP Terms: Odd (e.g., a-d, a, a+d); Even (e.g., a-3d, a-d, a+d, a+3d). 
? GP Terms: Odd (e.g., a/r, a, ar); Even (e.g., a/r
3
, a/r, ar, ar
3
). 
? HP Terms: Odd (e.g., 1/(a-d), 1/a, 1/(a+d)); Even (e.g., 1/(a-3d), 1/(a-d), 1/(a+d), 
1/(a+3d)). 
? Use partial fractions or telescoping for series sums, e.g., 1/(n
2
 - 1) = 1/(n-1) - 1/(n+1). 
Quick Reference Table 
Page 5


Sequences and Series Cheat Sheet 
(EduRev) 
Sequences 
Key De?nitions 
? Sequence: Ordered list of numbers (minimum 3 terms), e.g., {a
1
, a
2
, a
3
, ...}. 
? Finite Sequence: Fixed number of terms, e.g., {1, 3, 5, 7}. 
? In?nite Sequence: Continues inde?nitely, e.g., {1, 2, 3, ...}. 
? Rule: Formula for n-th term, e.g., T
n
 = 2n + 1 for {3, 5, 7, ...}. 
? Notation: T
n
 denotes n-th term, e.g., T
5
 for 5th term. 
Series 
Key De?nitions 
? Series: Sum of sequence terms, S
n
 = T
1
 + T
2
 + ... + T
n
. 
? Sigma Notation (S): Summation, e.g., S
n=1
k
 n = 1 + 2 + ... + k. 
? Pi Notation (?): Product, e.g., ?
n=1
k
 n = 1 × 2 × ... × k = k!. 
Sigma Properties 
? S
i=1
k
 a = ka (a is constant). 
? S
i=1
k
 (a
i
 ± b
i
) = S
i=1
k
 a
i
 ± S
i=1
k
 b
i
. 
Arithmetic Progression (AP) 
Formulas 
? n-th Term: T
n
 = a + (n-1)d, a = ?rst term, d = common difference. 
? n-th Term from End: T
m-n+1
 = T
m
 - (n-1)d, m = total terms. 
? Sum of n Terms: S
n
 = n/2 [2a + (n-1)d] = n/2 (a + T
n
). 
? Arithmetic Mean: AM = (a + b)/2 (two terms); AM = (a
1
 + ... + a
n
)/n (n terms). 
? n AMs between a and b: d = (b - a)/(n+1), terms = a + d, a + 2d, ..., a + nd. 
? Properties: 
? If T
m
 = n and T
n
 = m, then T
m+n
 = 0. 
? Sum of equidistant terms: T
k
 + T
n-k+1
 = a + T
n
. 
? Multiplying/dividing terms by constant C gives AP with difference Cd or 
d/C. 
? S
n
 = An
2
 + Bn, common difference = 2A. 
Geometric Progression (GP) 
Formulas 
? n-th Term: T
n
 = ar
n-1
, a = ?rst term, r = common ratio. 
? Property: T
n
 = v(T
n-1
 T
n+1
). 
? Sum of n Terms: S
n
 = a(1 - r
n
)/(1 - r) (r ? 1); S
n
 = na (r = 1). 
? In?nite GP Sum: S
8
 = a/(1 - r), for |r| < 1. 
? Geometric Mean: GM = v(ab) (two terms); GM = (a
1
 a
2
 ... a
n
)
1/n
 (n terms). 
? n GMs between a and b: r = (b/a)
1/(n+1)
, terms = ar, ar
2
, ..., ar
n
. 
Harmonic Progression (HP) 
Formulas 
? n-th Term: 1/a
n
 = 1/a + (n-1)d, where 1/a
n
 forms AP . 
? Harmonic Mean: HM = 2ab/(a + b) (two terms); HM = n/(1/a
1
 + ... + 1/a
n
) (n terms). 
? n HMs between a and b: 1/H
i
 = 1/a + i(a - b)/[(n+1)ab], i = 1, 2, ..., n. 
? Notes: No general sum formula; 0 cannot be a term. 
? Relation: AM = GM = HM, equality when numbers are equal. 
Arithmetic-Geometric Progression (AGP) 
Formulas 
? Form: ab, (a+d)br, (a+2d)br
2
, ... 
? Sum of n Terms: S
n
 = ab/(1-r) + dbr(1 - r
n-1
)/(1-r)
2
 - [a + (n-1)d]br
n
/(1-r) (r ? 1). 
? In?nite Sum: S
8
 = ab/(1-r) + dbr/(1-r)
2
, for |r| < 1. 
Special Series 
Formulas 
? Sum of n natural numbers: Sn = n(n+1)/2. 
? Sum of n odd numbers: S(2r-1) = n
2
. 
? Sum of n even numbers: S(2r) = n(n+1). 
? Sum of squares: Sn
2
 = n(n+1)(2n+1)/6. 
? Sum of cubes: Sn
3
 = [n(n+1)/2]
2
. 
? Special Sums: 
? S[1/(n(n+1))] = n/(n+1). 
? S[1/(n(n+1)(n+2))] = 1/4 - 1/[2(n+1)(n+2)]. 
? S[1/(n(n+1)(n+2)(n+3))] = 1/18 - 1/[3(n+1)(n+2)(n+3)], S
8
 = 1/18. 
AM, GM, HM Relations 
Formulas 
? Inequality: AM = GM = HM, equality when numbers are equal. 
? Geometric Relation: G
2
 = AH. 
? Quadratic Equation: For two numbers a, b: x
2
 - 2Ax + G
2
 = 0. 
? Cubic Equation: For three numbers a, b, c: x
3
 - 3Ax
2
 + (3G
3
/H)x - G
3
 = 0. 
? AM of m-th Power: (a
1
m
 + ... + a
n
m
)/n > (
m
v(a
1
 + ... + a
n
))
m
 if m ? [0,1]; reverse if m ? (0,1). 
Problem-Solving Tactics 
? Write out terms to spot patterns without simplifying. 
? AP Terms: Odd (e.g., a-d, a, a+d); Even (e.g., a-3d, a-d, a+d, a+3d). 
? GP Terms: Odd (e.g., a/r, a, ar); Even (e.g., a/r
3
, a/r, ar, ar
3
). 
? HP Terms: Odd (e.g., 1/(a-d), 1/a, 1/(a+d)); Even (e.g., 1/(a-3d), 1/(a-d), 1/(a+d), 
1/(a+3d)). 
? Use partial fractions or telescoping for series sums, e.g., 1/(n
2
 - 1) = 1/(n-1) - 1/(n+1). 
Quick Reference Table 
Concept Formula/Relation 
AP n-th Term T
n
 = a + (n-1)d 
AP Sum S
n
 = n/2 [2a + (n-1)d] 
GP n-th Term T
n
 = ar
n-1 
GP Sum S
n
 = a(1 - r
n
)/(1 - r), r ? 1 
In?nite GP Sum S
8
 = a/(1 - r), |r| < 1 
HP n-th Term 1/a
n
 = 1/a + (n-1)d 
Sum of n
2 
Sn
2
 = n(n+1)(2n+1)/6 
AM-GM-HM AM = GM = HM 
Solved Examples 
Example 1: AP n-th Term and Sum 
Problem: Find the 10th term and sum of the ?rst 10 terms of an AP with a = 3, d = 2.