Learn Sequences and Series with Free Lessons & Tips

1. Arithmetic progresssion: Arithmetic progression(AP) or arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.

An arithmetic progression is given by:

a, (a + d), (a + 2d), (a + 3d), ...

where a = the first term , d = the common difference

n^th term of an arithmetic progression

t_n = a + (n – 1)d

where t_n = n^th term, a = the first term, d = common difference

Number of terms of an arithmetic progression:

n=(l−a)d+1">n=(l−a)/d+1

where n = number of terms, a= the first term , l = last term, d= common differenc

Sum of first n terms in an arithmetic progression:

Sn=n2[ 2a+(n−1)d ] =n2(a+l)">Sn=n/2[ 2a+(n−1)d ] =n/2(a+l)

where a = the first term,
d= common difference,
l">l = t_n = n^th term = a + (n-1)d

2. Arithmetic Mean: If a, b, c are in AP, b is the Arithmetic Mean (AM) between a and c. In this case, b=12(a+c)">b=1/2(a+c)

b=12(a+c)">The Arithmetic Mean (AM) between two numbers a and b = 12(a+b)">1/2(a+b)

Solve most of the problems related to AP, the terms can be conveniently taken as:

3 terms: (a – d), a, (a +d)
4 terms: (a – 3d), (a – d), (a + d), (a +3d)
5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)

T_n = S_n - S_n-1

If each term of an AP is increased, decreased, multiplied or divided by the same non-zero constant, the resulting sequence also will be in AP.

In an AP, the sum of terms equidistant from beginning and end will be constant.

Geometric Progression(GP) or Geometric Sequence is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.

A geometric progression(GP) is given by a, ar, ar2, ar3,
where a = the first term, r = the common ratio

nth term of a geometric progression(GP):

tn=arn−1">tn=arn−1

where tn = nth term, a= the first term, r = common ratio, n = number of terms

Sum of first n terms in a Geometric Progression (GP):

Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sn=a(rn−1)/r−1 (if r>1)Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">a(1−rn)/1−r (if r<1)

Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sum of an infinite geometric progression(GP)

Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">S∞=a1−r  (if -1 < r < 1)">S∞=a/1−r  (if -1 < r < 1)

Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">where a= the first term, r = common ratio

Geometric MeanIf three non-zero numbers a, b, c are in GP, b is the Geometric Mean(GM) between a and c. In this case, b=ac">b=√ac

b=ac">The Geometric Mean(GM) between two numbers a and b = ab">√ab

(Note that if a and b are of opposite sign, their GM is not defined.)

Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Additional Notes on GP

To solve most of the problems related to GP, the terms of the GP can be conveniently taken as:
3 terms: ar">Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">a/r, a, ar
5 terms: ar2">ar2, ar">ar, a, ar, ar2.

If GM, AM and HM are the Geometric Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then,

GM² = AM × HM

Three numbers a, b and c are in AP if b=a+c2">b=a+c/2

Three non-zero numbers a, b and c are in HP if b=2aca+c">b=2ac/a+c

Three non-zero numbers a, b and c are in HP if a−bb−c=ac">a−b/b−c=ac

a−bb−c=ac">Let A, G and H be the AM, GM and HM between two distinct positive numbers. Then

(1) A > G > H
(2) A, G and H are in GP

a−bb−c=ac">If a series is both an AP and GP, all terms of the series will be equal. In other words, it will be a constant sequence.

a−bb−c=ac">Power Series : Important formulas

1+1+1+â?¯ n terms">1+1+1+â?¯ n terms

a−bb−c=ac">=∑1=n">=∑1=n

a−bb−c=ac"> 

1+2+3+â?¯+n">1+2+3+â?¯+n

a−bb−c=ac">=∑n=n(n+1)2">=∑n=n(n+1)2

a−bb−c=ac"> 

12+22+32+â?¯+n2">12+22+32+â?¯+n2

a−bb−c=ac">=∑n2=n(n+1)(2n+1)6">=∑n2=n(n+1)(2n+1)6

read less