Q : 2      Find the sums given below :

             (i)   \small 7+10\frac{1}{2}+14+...+84

Answers (1)

Given AP is 
\small 7+10\frac{1}{2}+14+...+84
We first need to find the number of terms
Here, a = 7 \ and \ a_n = 84
And
d = a_2-a_1=\frac{21}{2}-7= \frac{21-14}{2}= \frac{7}{2}
Let suppose there are n terms in the AP
Now, we know that
a_n = a+(n-1)d
\Rightarrow 84 = 7 + (n-1)\frac{7}{2}
\Rightarrow \frac{7n}{2}= 77+\frac{7}{2}
\Rightarrow n = 23
Now, we know that 
S = \frac{n}{2}\left \{ 2a+(n-1)d \right \}
\Rightarrow S = \frac{23}{2}\left \{ 2\times7 +(23-1)(\frac{7}{2})\right \}
\Rightarrow S = \frac{23}{2}\left \{ 14 +77\right \}
\Rightarrow S = \frac{23}{2}\left \{ 91\right \}
\Rightarrow S =\frac{2093}{2}=1046\frac{1}{2}
Therefore, the sum of AP \small 7+10\frac{1}{2}+14+...+84   is 1046\frac{1}{2}

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