Question

Find x where x=11/3+11/8+11/15+11/24+11/35+11/48+
11/63+11/80+11/99
Without using LCM

Answer 1

Given : x=11/3+11/8+11/15+11/24+11/35+11/48+ 11/63+11/80+11/99

To Find :  Value of x  

Solution:

11/3  + 11/8  + 11/15  +  11/24   + 11/35  + 11/48   +  11/63  + 11/80  + 11/99

Taking 11 common

= 11 ( 1/3  + 1/8  +  1/15  + 1/24  + 1/35  + 1/48  + 1/63  + 1/80  + 1/99)

multiplying and dividing by 2

= (11 /2 )( 2/3  + 2/8  + 2/15  +  2/24  + 2/35  + 2/48  +2/63  + 2/80  + 2/99)

rewriting terms  like 3 = 1 * 3 , 8 = 2 * 4 and so on till 99 = 9 * 11

= (11 /2 )( 2/(1 * 3)  + 2/(2 * 4)  + 2/(3 * 5)  +  2/(4 *6)  + 2/(5 * 7)  + 2/(6 *8)  +2/(7 * 9)  + 2/(8 * 10)  + 2/(9 * 11))

rewriting terms  2 = 3 - 1  , 2 = 4 -2  and so on till 2 = 11 - 9

=  (11 /2 )( (3 - 1)/(1 * 3)  + (4-2)/(2 * 4)  + (5-3)/(3 * 5)  +  (6-4)/(4 *6)  + (7-5)/(5 * 7)  + (8-6)/(6 *8)  +(9-7)/(7 * 9)  + (10-8)/(8 * 10)  + (11-9)/(9 * 11))

Splitting numerator terms

= (11/2) ( 1 - 1/3  + 1/2 - 1/4  + 1/3 - 1/5 + 1/4 - 1/6 + 1/5 - 1/7 + 1/6 - 1/8  + 1/7 - 1/9 + 1/8 - 1/10 + 1/9 - 1/11)

cancelling similar  terms with opposite sign

= (11/2) ( 1 + 1/2  -1/10  -  1/11)

=(11/2) ( 3/2  - 21/110)

= 33/4 -  21/20

= 8.25 - 1.05

= 7.2

Learn More:

Find x where x=11/3+11/8+11/15+11/24+11/35+11/48+ 11/63+11/80+11/99

https://brainly.in/question/10011880

Answer 2

SOLUTION

TO DETERMINE

The value of x when

$\displaystyle\sf{x = \frac{11}{3} + \frac{11}{8} + \frac{11}{15} + \frac{11}{24} + \frac{11}{35} + \frac{11}{48} + \frac{11}{63} + \frac{11}{80} + \frac{11}{99}}$

EVALUATION

$\displaystyle\sf{x = \frac{11}{3} + \frac{11}{8} + \frac{11}{15} + \frac{11}{24} + \frac{11}{35} + \frac{11}{48} + \frac{11}{63} + \frac{11}{80} + \frac{11}{99}}$

$\displaystyle\sf{ \implies \: x = \frac{11}{ {2}^{2} - 1 } + \frac{11}{ {3}^{2} - 1} + \frac{11}{ {4}^{2} - 1 } + \frac{11}{ {5}^{2} - 1} + \frac{11}{ {6}^{2} - 1} + \frac{11}{ {7}^{2} - 1 } + \frac{11}{ {8}^{2} - 1 } + \frac{11}{ {9}^{2} - 1 } + \frac{11}{ {10}^{2} - 1}}$

$\displaystyle \sf{ \implies \: x = \sum\limits_{n=2}^{10} \: \frac{11}{ {n}^{2} - 1 } }$

$\displaystyle \sf{ \implies \: x =11 \: \sum\limits_{n=2}^{10} \: \frac{1}{ (n - 1)(n + 1) } }$

$\displaystyle \sf{ \implies \: x = \frac{11}{2} \: \sum\limits_{n=2}^{10} \: \frac{(n + 1) - (n - 1) }{ (n - 1)(n + 1) } }$

$\displaystyle \sf{ \implies \: x = \frac{11}{2} \bigg[ \: \sum\limits_{n=2}^{10} \: \frac{1 }{ (n - 1) } -\: \frac{1 }{ (n + 1) } \bigg] }$

$\displaystyle \sf{ \implies \: x = \frac{11}{2} \bigg[ \: \sum\limits_{n=2}^{10} \: \frac{1 }{ (n - 1) } - \frac{1}{n} + \frac{1}{n} - \frac{1 }{ (n + 1) } \bigg] }$

$\displaystyle \sf{ \implies \: x = \frac{11}{2} \bigg[ \: \sum\limits_{n=2}^{10} \: \frac{1 }{ (n - 1) } - \frac{1}{n} \bigg] +\frac{11}{2} \bigg[ \: \sum\limits_{n=2}^{10} \: \frac{1}{n} - \frac{1 }{ (n + 1) } \bigg] }$

$\displaystyle \sf{ \implies \: x = \frac{11}{2} \bigg[ \: 1 - \frac{1}{10} + \frac{1}{2} - \frac{1}{11} \bigg] }$

$\displaystyle \sf{ \implies \: x = \frac{11}{2} \times \frac{144}{110} }$

$\displaystyle \sf{ \implies \: x = 7.2 }$

FINAL ANSWER

Hence the required value of x = 7.2

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