Question

. Find the smallest term of geometric progression 3,5,25/3
that exceeds 100.​

Answer 1

Given : geometric progression 3, 5, 25/3    

To Find : smallest term of geometric progression  that exceeds 100.​

Solution:

3 , 5   , 25/3

a = 3

r =  5/3  

nth term  = 3 (5/3)ⁿ⁻¹   >  100

3 (5/3)ⁿ⁻¹   >  100

=> (5/3)ⁿ⁻¹ > 100/3

=> (5/3)ⁿ⁻¹ > 20 * 5 /3

=> (5/3)ⁿ⁻² > 20

n - 2 = b

=> (5/3)ᵇ > 20

=> 5ᵇ > 20 * 3ᵇ

b = 5

3125 < 4860

b = 6

15625 > 14580

b = 6

n - 2 = b

=> n = 8

3 (5/3)ⁿ⁻¹ = 3 (5/3)⁷  =  78125/729  

78125/729   is the smallest term of geometric progression 3,5,25/3

that exceeds 100.

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Answer 2

SOLUTION

TO DETERMINE

The smallest term of geometric progression

$\displaystyle \sf{3,5 ,\frac{25}{3} , ......\: }$

that exceeds 100.

EVALUATION

In the Geometric Progression

First term = a = 3

$\displaystyle \sf{ Common \: Ratio = r =\: \: \frac{5}{3} }$

$\sf{Let \: T_{n} \: be \: the \: smallest \: term \: that \: exceeds \: 100 \: }$

$\displaystyle \sf{T_{n} = 3 \times { \bigg( \frac{5}{3} \bigg)}^{n - 1} }$

So by the given condition

$\displaystyle \sf{ 3 \times { \bigg( \frac{5}{3} \bigg)}^{n - 1} > 100 }$

$\implies\displaystyle \sf{ { \bigg( \frac{5}{3} \bigg)}^{n - 1} > \frac{100}{3} }$

Taking logarithm in both sides we get

$\displaystyle \sf{ \log{ \bigg( \frac{5}{3} \bigg)}^{n - 1} > \log \bigg( \frac{100}{3} \bigg)}$

$\implies \displaystyle \sf{ (n - 1)\log{ \bigg( \frac{5}{3} \bigg)} > \log \bigg( \frac{100}{3} \bigg)}$

$\implies \displaystyle \sf{ (n - 1) \times 0.222 > 1.523}$

$\implies \displaystyle \sf{ (n - 1) > \frac{1.523}{0.222} }$

$\implies \displaystyle \sf{ (n - 1) > 6.86 }$

$\implies \displaystyle \sf{ n > 7.86 }$

Now the smallest natural number greater than 7.86 is 8

Hence 8 th term is the smallest term of the geometric progression that exceeds 100 and the term is

$\displaystyle \sf{T_{8} = 3 \times { \bigg( \frac{5}{3} \bigg)}^{8 - 1} }$

$\displaystyle \sf{= 3 \times { \bigg( \frac{5}{3} \bigg)}^{7} }$

$= \displaystyle \sf{ \frac{78125}{729} }$

$\displaystyle \sf{ = 107 \: \frac{122}{729} }$

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