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Lesson Posted on 05 Feb Sequences and Series

Arithmetic Progression

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),... read more

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=(la)/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+(n1)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.

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Lesson Posted on 05 Feb Sequences and Series

Geometric Progression

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 rationth term of a geometric progression(GP):tn=arn−1">tn=arn−1where... read more

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.

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Lesson Posted on 05 Feb 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/2Three non-zero numbers a, b and c are in HP if b=2aca+c">b=2ac/a+cThree non-zero numbers a, b and c are... read more

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">ab/bc=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+? n terms">1+1+1+â?¯ n terms

a−bb−c=ac">=∑1=n">=1=n

 

a−bb−c=ac"> 

1+2+3+?+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+?+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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Answered on 28/11/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

Tapas Bhattacharya

Tutor

tn = 3 x 4^(n-1).
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Answered on 29/11/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

Find the four arithmetic means between 16 and 91?

Tapas Bhattacharya

Tutor

Here, the arithmetic means are p, q, r, and s. So, 16, p, q, r, s, 91, ..... are in A. P. in which t1 = 16, and t6 = 91. Solving , we can get, a = 16, d = 15. so, p = 16 + 15 = 31, q = 31+15 = 46, r = 46 + 15 = 61, s = 61 + 15 = 76. So, the arithmetic means are: 31, 46, 61, 76.
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Answered on 29/11/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

Find the next four terms of arithmetic sequence: 12, 16, 20,....?

Tapas Bhattacharya

Tutor

24, 28, 32, 36.
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Answered on 03/12/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

Find the sum of series: 18 + 22 + 26 + 30 +..... + 50?

Sarvajeet Kumar

An Experienced Trainer

306.
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Answered on 03/12/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

What is the formula for arithmetic mean in sequence and series?

Sarvajeet Kumar

An Experienced Trainer

a, b, c are in A. P. b=(a+c)/2.
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Answered on 19/11/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

Find the 8th term of a geometric sequence for which a = -3 and r = -2?

Richa Kumari

384
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Answered on 29/11/2016 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC) Sequences and Series

Find the next two terms of each geometric sequence: 20, 30, 45, ....?

Tapas Bhattacharya

Tutor

Here, a = 20 and r = 30/20 = 45/30 = 3/2 = 1.5. So, next two terms are, 45 x 1.5 = 67.5 and 67.5 x 1.5 = 101.25.
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