x b a xb-a

Algebra Level 3

1 9 , 1 25 , 1 43 , 1 63 , 1 65 , 1 85 \frac{1}{9},{\frac{1}{25}},{\frac{1}{43}},{\frac{1}{63}},{\frac{1}{65}},{\frac{1}{85}}{\dots}

Find the 22nd term of the sequence above.

1 303 \frac{1}{303} 1 347 \frac{1}{347} 1 325 \frac{1}{325} 1 369 \frac{1}{369} 1 25 \frac{1}{25} 1 391 \frac{1}{391}

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1 solution

Matin Naseri
Aug 31, 2018

1 9 \frac{1}{9} can be expressed as 1 14 n 5 \boxed{\frac{1}{14n-5}}

1 25 \frac{1}{25} can be expressed as 1 15 n 5 \boxed{\frac{1}{15n-5}}

1 43 \frac{1}{43} can be expressed as 1 16 n 5 \boxed{\frac{1}{16n-5}}

1 63 \frac{1}{63} can be expressed as 1 17 n 5 \boxed{\frac{1}{17n-5}}

1 65 \frac{1}{65} can be expressed as 1 14 n 5 \boxed{\frac{1}{14n-5}}

1 85 \frac{1}{85} can be expressed as 1 15 n 5 \boxed{\frac{1}{15n-5}}

\vdots

22 4 ( m o d 2 ) 22{\equiv}4{\pmod{2}} and we have 1 14 n 5 \frac{1}{14n-5} , 1 15 n 5 \boxed{\frac{1}{15n-5}} , 1 16 n 5 \frac{1}{16n-5} , 1 17 n 5 \frac{1}{17n-5} \dots

Then 2 2 n d 22^{nd} expressed as 1 15 n 5 \frac{1}{15n-5} .

1 22 × 15 5 1 330 5 = 1 325 \frac{1}{22×15-5}{\implies}~{\frac{1}{330-5}}={\boxed{\frac{1}{325}}}

Hence the answer as 1 325 \color{#3D99F6}{\boxed{{\frac{1}{325}}}}

Wow, Amazing solution. I want really like your problem but I can't.

Tom Clancy - 2 years, 9 months ago

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