Question

IF 1/1!9!+1/3!7!+1/5!5!+1/7!3!+1/9!=2powern/10! then n=?​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO DETERMINE

The value of n when

$\displaystyle \sf{ \frac{1}{1 \: ! \: 9 \: ! } + \frac{1}{3\: ! \: 7 \: ! } + \frac{1}{5 \: ! \: 5 \: ! } + \frac{1}{7 \: ! \: 3 \: ! } + \frac{1}{1 \: ! \: 9 \: ! } \: = \frac{ {2}^{n} }{1 0\: ! } }$

CALCULATION

Here

$\displaystyle \sf{ \frac{1}{1 \: ! \: 9 \: ! } + \frac{1}{3\: ! \: 7 \: ! } + \frac{1}{5 \: ! \: 5 \: ! } + \frac{1}{7 \: ! \: 3 \: ! } + \frac{1}{1 \: ! \: 9 \: ! } \: = \frac{ {2}^{n} }{1 0\: ! } }$

$\implies \displaystyle \sf{ \frac{10 \: !}{1 \: ! \: 9 \: ! } + \frac{10 \: !}{3\: ! \: 7 \: ! } + \frac{10 \: !}{5 \: ! \: 5 \: ! } + \frac{10 \: !}{7 \: ! \: 3 \: ! } + \frac{10 \: !}{1 \: ! \: 9 \: ! } \: = {2}^{n} }$

$\implies \displaystyle \sf{ \frac{10 \times 9 \: !}{\: 9 \: ! } + \frac{10 \times 9 \times 8}{3 \times 2} + \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 } + \frac{10 \times 9 \times 8}{3 \times 2 } + \frac{10 \times9 \: ! }{ 9 \: ! } \: = {2}^{n} }$

$\implies \: \displaystyle \sf{10 + 120 + 252 + 120 + 10 = {2}^{n} }$

$\implies \: \displaystyle \sf{ {2}^{n} = 512 }$

$\implies \: \displaystyle \sf{ {2}^{n} = {2}^{9} }$

$\implies \: \displaystyle \sf{ n = 9 }$

RESULT

Hence the required value of n is 9

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Answer 2

Step-by-step explanation:

\displaystyle\huge\red{\underline{\underline{Solution}}}

Solution

TO DETERMINE

The value of n when

\displaystyle \sf{ \frac{1}{1 \: ! \: 9 \: ! } + \frac{1}{3\: ! \: 7 \: ! } + \frac{1}{5 \: ! \: 5 \: ! } + \frac{1}{7 \: ! \: 3 \: ! } + \frac{1}{1 \: ! \: 9 \: ! } \: = \frac{ {2}^{n} }{1 0\: ! } }

1!9 3!7

CALCULATION

Here

\displaystyle \sf{ \frac{1}{1 \: ! \: 9 \: ! } + \frac{1}{3\: ! \: 7 \: ! } + \frac{1}{5 \: ! \: 5 \: ! } + \frac{1}{7 \: ! \: 3 \: ! } + \frac{1}{1 \: ! \: 9 \: ! } \: = \frac{ {2}^{n} }{1 0\: ! } }

1!9!

1

+

3!7!

1

+

5!5!

1

+

7!3!

1

+

1!9!

1

=

10!

2

n

\implies \displaystyle \sf{ \frac{10 \: !}{1 \: ! \: 9 \: ! } + \frac{10 \: !}{3\: ! \: 7 \: ! } + \frac{10 \: !}{5 \: ! \: 5 \: ! } + \frac{10 \: !}{7 \: ! \: 3 \: ! } + \frac{10 \: !}{1 \: ! \: 9 \: ! } \: = {2}^{n} }⟹

1!9!

10!

+

3!7!

10!

+

5!5!

10!

+

7!3!

10!

+

1!9!

10!

=2

n

\implies \displaystyle \sf{ \frac{10 \times 9 \: !}{\: 9 \: ! } + \frac{10 \times 9 \times 8}{3 \times 2} + \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 } + \frac{10 \times 9 \times 8}{3 \times 2 } + \frac{10 \times9 \: ! }{ 9 \: ! } \: = {2}^{n} }⟹

9!

10×9!

+

3×2

10×9×8

+

5×4×3×2

10×9×8×7×6

+

3×2

10×9×8

+

9!

10×9!

=2

n

\implies \: \displaystyle \sf{10 + 120 + 252 + 120 + 10 = {2}^{n} }⟹10+120+252+120+10=2

n

\implies \: \displaystyle \sf{ {2}^{n} = 512 }⟹2

n

=512

\implies \: \displaystyle \sf{ {2}^{n} = {2}^{9} }⟹2

n

=2

9

\implies \: \displaystyle \sf{ n = 9 }⟹n=9

RESULT

Hence the required value of n is 9

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

The highest exponent of 24 in 20 !

https://brainly.in/question/21368441