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Lesson Posted on 30 May Progressions

Geometric Progression : Basics

Hemant P.

1. I have taught at various prestigious institutes of repute in India and abroad. 2. Worked at (i)...

Geometric Mean (GM) The geometric mean is similar to the arithmetic mean, but it uses the property of multiplication, to find the mean between any two numbers. GM of two numbers a and b may be defined as G = √(ab)For example, GM of 2 and 3 is √6.We can find a general formula for inserting... read more

Geometric Mean (GM)

 

The geometric mean is similar to the arithmetic mean, but it uses the property of multiplication, to find the mean between any two numbers.

GM of two numbers a and b may be defined as G = √(ab)

For example, GM of 2 and 3 is √6.

We can find a general formula for inserting a GM between any two arbitrary numbers a and b.

 

 If G is the geometric mean between a and b, then by definition of GP a,  G,  b are in GP.

Or G2=a.b

 

We can also find a formula for inserting n GM in between two numbers a and b.

Let  G1,  G2,  G3  ….  Gn be then GMs between a and b.

Now a,   G1,  G2, G3…. Gn,  b are in GP.

By definition of the last term of a GP  b = (n+2)th term of GP.

If R=common ratio, then by definition of GP,  b = a.Rn+1      

Now G1=a?R

       G2=a?R2

         G3=a?R3

   and Gn=a?Rn

 

We will take a few examples of GM too.

 

 

Find the GM of the following and insert n geometric means between them.

 

    e and 1
    7 and tan x
    log x and sec-1x

 

Solution:

(i) The GM between e and 1 is e. To insert n GMs, we use the definition of geometric progressions.

Let  G1,   G2,  G3,  …  Gn,  be n  geometric means inserted between e and 1.

Also, let common ratio be R.

 By the definition of GP    = (n+2)th term of the GP.

 1 =eRn+1

Or R= (1/e)1/(n+1)

Hence   G1 = e⋅R = e⋅(1/e)1/(n+1)

            G2=e⋅R2=e⋅(1/e)2/(n+1)

            G3=e⋅R=e⋅(1/e)3/(n+1)

            and  Gn=e⋅R=e⋅(1/e)n/(n+1)

 

 

(ii)   Proceeding as before the GM  of 7 and tanx is √(7tanx).

Let G1, G2, G3,… Gn,  be n  geometric means inserted between 7 and tanx.Also, let common ratio be R.

To get the n geometric means, we proceed as before.

Here tanx=(n+2)th term.

 tanx=7⋅Rn+1

Or Rn+1= tanx/7

R = (tanx/7)1/n+1

 G1 = 7R = 7⋅(tanx/7)1/n+1

    G2  =  7R2   = 7⋅(tanx/7)2/n+1

    G3 = 7R3 = 7⋅(tanx/7)3/n+1

 

and   Gn=7Rn=7(tanx/7)n/n+1

 

(iii)  Proceeding exactly as before, the GM of logx and sec-1x will be √(logx⋅sec-1x)

 Let   G1,  G2,  G3, …  Gn,  be n  geometric means inserted between logx and sec-1x.

Also, let common ratio be R.

    

To insert n GMs between logx and sec-1x, we take the sequence as

    logx, G1, G2, G3,… Gn, sec-1x.

 Gn = (n+2)th term =  sec-1x = logx.Rn+1

 R= (sec-1x/logx)1/(n+1)

Now as before G1=logx⋅R=logx⋅(sec-1x/logx)1/(n+1)

                       G2=logx⋅R2=logx⋅(sec-1x/logx)2/(n+1)

                       G3=logx⋅R3=logx⋅(sec-1x/logx)3/(n+1)

           and Gn=logx⋅Rn=logx⋅(sec-1x/logx)n/(n+1)

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Answered on 30/11/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

Tapas Bhattacharya

Tutor

The first term, t1=a = 7. The common difference, b=13 - 7=6. Last term or the nth term, tn=a + (n-1)b = 205. So, we have, 7 + (n - 1)6 = 205 --> (n - 1) = (205 - 7)/6 = 33 --> n = 34. Answer: Number of terms is 34.
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

Find the A.P whose 10th term is 5 and 18th term is 77?

Tapas Bhattacharya

Tutor

Let the first term be a and the common difference be b. So, the 10th term: t10 = a + 9b = 5. and, the 18th term: t18 = a + 17b = 77. So, from t17 - t10 we get: 8b = 77 - 5 = 72. This gives: b = 9. From t10 we get: a = 5 - 9b = 5 - 9x9 = -76. So, a = -76, b = 9 and the A.P is:- -76, -67, -58, -49,... read more
Let the first term be a and the common difference be b. So, the 10th term: t10 = a + 9b = 5. and, the 18th term: t18 = a + 17b = 77. So, from t17 - t10 we get: 8b = 77 - 5 = 72. This gives: b = 9. From t10 we get: a = 5 - 9b = 5 - 9x9 = -76. So, a = -76, b = 9 and the A.P is:- -76, -67, -58, -49, ..... (Answer). read less
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

In a certain A.P the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term?

Tapas Bhattacharya

Tutor

Let the first term be a and the common difference be b. So, by the given condition:- t24=2 x t10 --> a + 23 b = 2 ( a + 9b) = 2a + 18b. This gives: a = 5b. So, the 72nd term: t72 = a +71b = 5b + 71b = 76b. and the 34th term: t34 = a + 33b = 5b + 33b = 38b = 76b/2 = t72/2. So, the 72nd term is twice... read more
Let the first term be a and the common difference be b. So, by the given condition:- t24=2 x t10 --> a + 23 b = 2 ( a + 9b) = 2a + 18b. This gives: a = 5b. So, the 72nd term: t72 = a +71b = 5b + 71b = 76b. and the 34th term: t34 = a + 33b = 5b + 33b = 38b = 76b/2 = t72/2. So, the 72nd term is twice the 34th term. (Proved) read less
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

The sum of the first six terms of an A.p is zero and the fourth term is 2. Find the sum of its first 30 terms?

Tapas Bhattacharya

Tutor

Let the first term be (a - 5b) and the common difference be 2b. So the six terms are: (a - 5b), (a - 3b), (a - b), (a + b), (a + 3b), (a + 5b). By the given conditions:- Sum of six terms = (a - 5b) + (a - 3b) + (a - b) + (a + b) + (a + 3b) + (a + 5b) = 6a = 0. So, a = 0. The fourth term = a + b = 2... read more
Let the first term be (a - 5b) and the common difference be 2b. So the six terms are: (a - 5b), (a - 3b), (a - b), (a + b), (a + 3b), (a + 5b). By the given conditions:- Sum of six terms=(a - 5b) + (a - 3b) + (a - b) + (a + b) + (a + 3b) + (a + 5b) = 6a=0. So, a=0. The fourth term=a + b=2 --> 0 + b=2 --> b = 2. So, for the A.P. the first term x = (a - 5b) = -10. The common difference, y = 2b = 4. Sum of 1st 30 terms, S30 = (n/2)[2x + (n-1)y] = 15 x [ -20 + 29 x 4] = 1440. (Answer) read less
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

An A.P consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term?

Tapas Bhattacharya

Tutor

Let the first term be a and the common difference be b. So, t1 = a = 7 and t60 = a + 59b = 125. From above, we get:- t60 - t1 = 59b = 125 - 7 = 118. So, b = 118/59 = 2. Hence, the 32nd term, t32 = a + 31b = =7 + 31x2 = 69. (Answer)
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers?

Tapas Bhattacharya

Tutor

Let the 3 terms are (a - b), a, and (a + b). So, (a - b) + a + ( a + b) = 27=> 3a=27 => a=9. Now the product is 648. So, (9 - b) x 9 x (9 + b) = 648=> (81 - b^2) = 648/9 = 72=> b^2 = 81 - 72=9 => b = =3 or b = -3. For, b = 3, the numbers are: (9 - 3), 9, (9 + 3) = 6, 9, 12 or (9 - (-3)), 9... read more
Let the 3 terms are (a - b), a, and (a + b). So, (a - b) + a + ( a + b) = 27=> 3a=27 => a=9. Now the product is 648. So, (9 - b) x 9 x (9 + b) = 648=> (81 - b^2) = 648/9 = 72=> b^2 = 81 - 72=9 => b = =3 or b = -3. For, b = 3, the numbers are: (9 - 3), 9, (9 + 3) = 6, 9, 12 or (9 - (-3)), 9 , (9 + (-3)) = 12, 9, 6 The three numbers are 6, 9, 12 or 12, 9, 6 (Answer) read less
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

Write the first four terms of the A.p, when the first term a and the common difference d are given: a = 10, d = 10?

Tapas Bhattacharya

Tutor

The first four terms are 10, 10 + 10, 10 + 2x10, and 10 + 3x10. So, the terms are: 10, 20, 30, 40.
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

For the given A.P write the first term and the common difference: 3, 1, -1, -3, ?..?

Tapas Bhattacharya

Tutor

The first term is 3. The common difference = 1-3 = -2.
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Answered on 01/12/2016 Tuition/Class IX-X Tuition Tuition/Class IX-X Tuition/Mathematics Progressions

If the 3rd and the 9th terms of an A.P are 4 and -8 respectively, which term of this A.P is zero?

Tapas Bhattacharya

Tutor

Let the first term and the common difference are a and b. So, the third term: t3=a + 2b=4. and, the 9th term: t9=a + 8b=-8. So, t9 - t6=6b = -8 - 4=-12 --> b = - 2. So, from the expression of t3, we get: a = 4 - 2b = 4 + 4 = 8. Let us consider that the nth term is zero. So: tn... read more
Let the first term and the common difference are a and b. So, the third term: t3=a + 2b=4. and, the 9th term: t9=a + 8b=-8. So, t9 - t6=6b = -8 - 4=-12 --> b=- 2. So, from the expression of t3, we get: a=4 - 2b=4 + 4=8. Let us consider that the nth term is zero. So: tn=a + (n-1)b = 0 --> 8 + (n-1)(-2) = 0 --> 8 -2n +2 = 0 --> 2n=10 --> n = 5. Answer: The 5th term of the series is zero. read less
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