What is 12+16+112+120+130+142+156?

Question

What is  \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56} ?

A.  \frac{5}{7} 
 
B.  \frac{5}{8} 
 
C.  \frac{7}{8 }  
 
D.  \frac{15}{56}  
 
E.  \frac{23}{56}

Kinshook · Accepted Answer

Asked: What is  \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56} ?

\frac{1}{n(n+1)} = \frac{1}{n }- \frac{1}{n+1}

\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}

(\frac{1}{1} - \frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4}) + ... + (\frac{1}{7}-\frac{1}{8})

1 - \frac{1}{8} = \frac{7}{8}

IMO C

MS2020 · Answer

I solved this by looking at the answer options.

Since S  = 1/2  + 1/6 + 1/12 + ... + 1/56, 
the answer should be greater than 1/2.

This eliminates options D (15/56 ~= 0.26) and E (23/56 ~= 0.41)

Add 1/2 + 1/6 + 1/12
This gives us 3/4. So now we can eliminate answer choices A(5/7 ~= 0.71) B (5/8 = 0.625) which are both less than 0.75.

Thus the answer is C

yashikaaggarwal · Answer

=> 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 
=> 1/2 + {1/6 + 1/12} + {1/20 + 1/30} + {1/42 + 1/56}
=> {1/2 + 1/4} + {1/12 + 1/24}
=> 3/4 + 3/24
=> 18+3/24
=> 21/24
=> 7/8

Answer is C

Posted from my mobile device

rajatchopra1994 · Answer

Explanation:
Tn can be stated as 1/{n*(n+1)}
i.e. Tn=1/{n*(n+1)}={(n+1)-n}/{n*(n+1)}
Tn=1/n- 1/n+1
So, Sn=1- 1/n+1

here, n=7
So,

Sn= 1- 1/(7+1) = 1 - 1/8 = 7/8

IMO-C

MathRevolution · Answer

=>

Remember that  \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}.

\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}

=\frac{1}{1∙2}+\frac{1}{2∙3}+\frac{1}{3∙4}+\frac{1}{4∙5}+\frac{1}{5∙6}+\frac{1}{6∙7}+\frac{1}{7∙8}

=(\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+(\frac{1}{5}-\frac{1}{6})+(\frac{1}{6}-\frac{1}{7})+(\frac{1}{7}-\frac{1}{8})

=  1-\frac{1}{8}  (because the last term of each bracket cancels out the first term of the next bracket, leaving us only the very first and very last terms)

=  \frac{7}{8}

Therefore, C is the correct answer.
Answer: C

BlueRocketAsh · Answer

1/2,1/6,1/12,1/20,1/30,1/42,1/56
Common term for this series is 1/n(n+1)
By partial fractions 1/n(n+1) = 1/n - 1(n+1)

Then the series becomes;
(1/1 - 1/2) , (1/2 - 1/3) , (1/3 + 1/4) ,............, (1/7 - 1/8)

Sum of the series;
(1/1 - 1/2) + (1/2 - 1/3) + (1/3 + 1/4) +............+ (1/7 - 1/8)
= 1 - 1/8
=7/8

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What is 12+16+112+120+130+142+156?

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