## Find the Sum of a Sequence

For students of class 11-12 (age 16+)
Moon
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### Find the Sum of a Sequence

The sequence $a_1, a_2,\cdots , a_{98}$ satisfies $a_{n+1} =a_n + 1$ for $n = 1, 2, \cdots , 97$ and has sum $137$.
$a_2 + a_4 + a_6 + \cdots + a_{98}=?$
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Labib
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### Re: Find the Sum of a Sequence

Here's how I've worked on it!!

Soln::
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MathWasheefSci
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Joined: Tue Dec 07, 2010 12:20 pm
Location: Tejgaon, Dhaka, Bangladesh-1215

### Re: Find the Sum of a Sequence

I have another technique. Given that $a_{1}+a_{2}+a_{3}+.......+a_{98}=137$ $\Leftrightarrow S+a_{1}+a_{3}+a_{5}+.......+a_{97}=137\; ;[let\, \; a_{2}+a_{4}+.....+a_{98}=S]$$\Leftrightarrow S+a_{1}+(a_{2}+1)+(a_{4}+1)+.......+(a_{96}+1)=137\; \: ;[\because a_{n+1}=a_{n}+1]$$\Leftrightarrow 2S+a_{1}-a_{98}+48=137\; \; ;[\because \textrm{There are 48 terms}]$
Now $\begin{matrix} &a_{2} &-a_{1}=1 \\ &a_{3} &-a_{2}=1 \\ &a_{4} &-a_{3}=1 \\ &.... &.... \\ &a_{98} &-a_{97}=1 \end{matrix}$ which sum up to $a_{98}-a_{1}=97$
$\therefore 2S-97+48=137$
$\therefore S=93$

Tahsin24
Posts: 21
Joined: Tue Dec 07, 2010 6:13 pm

### Re: Find the Sum of a Sequence

the question says that the summation is 137 for n=1,2,3,...........,97. Not 98