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Lesson Posted on 05/02/2018 Learn Sequences and Series
Raj Kumar
I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...
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
nth term of an arithmetic progression
tn = a + (n – 1)d
where tn = nth 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 = tn = nth 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)
Tn = Sn - Sn-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.
Lesson Posted on 05/02/2018 Learn Sequences and Series
Raj Kumar
I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...
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.
Lesson Posted on 05/02/2018 Learn Sequences and Series
Relationship between A.M, H.M & G.M
Raj Kumar
I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...
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
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