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Find the sums given below:
(i) $7 + 10\frac{1}{2} + 14 + … + 84$
(ii) $34 + 32 + 30 + … + 10$
(iii) $-5 + (-8) + (-11) + ….. + (-230)$
To do:
We have to find the given sums.
Solution:
(i) $a=7, d=10\frac{1}{2}-\frac{7}{1}=\frac{21}{2}-\frac{7}{1}=\frac{21-14}{2}=\frac{7}{2}, a_n=84$
We know that,
$a_n=a+(n-1)d$
$S_{n}=\frac{n}{2}[a+l]$
$a_{n}=a+(n-1) d$
$84=7+(n-1) \frac{7}{2}$
$84-7=(n-1) \frac{7}{2}$
$77(\frac{2}{7})=n-1$
$11(2)=n-1$
$22+1=n$
$n=23$
Therefore,
$S_{n}=\frac{n}{2}[a+l]$
$S_{23}=\frac{23}{2}[7+84]$
$=\frac{23}{2} \times 91$
$=\frac{2093}{2}$
$=1046\frac{1}{2}$
(ii) $a=34, d=32-34=-2, a_n=10$
We know that,
$a_n=l=a+(n-1)d$
$S_{n}=\frac{n}{2}[a+l]$
$a_{n}=a+(n-1) d$
$10=34+(n-1)(-2)$
$10-34=(n-1)(-2)$
$\frac{-24}{-2}=n-1$
$n-1=12$
$n=12+1$
$n=13$
Therefore,
$S_{13}=\frac{13}{2}[34+10]$
$=\frac{13}{2}(44)$
$=13 \times 22$
$=286$
(iii) Here,
$(-5)+(-8)+(-11)+\ldots+(-230)$ is in A.P.
$a=-5, d=-8-(-5)=-8+5=-3$ and $l=-230$
We know that,
$a_{n}=a+(n-1) d$
$\Rightarrow-230=-5+(n-1)(-3)$
$-230=-5-3 n+3$
$3 n=-5+3+230=228$
$n=\frac{228}{3}=76$
$S_{n}=\frac{n}{2}[a+l]$
$=\frac{76}{2}[-5+(-230)]$
$=38(-5-230)$
$=38(-235)$
$=-8930$
Therefore, the sum of the given sequence is $-8930$.