Gujarat BoardEnglish MediumSTD 10MathsArithmetic Progressions3 Marks
Question
Find the sum of $-5 + (-8) + (-11) + …. + (-230).$
✓
Answer
Let $S_n=-5+(-8)+(-11)+\ldots .+(-230)$
Clearly, the terms of the sum form an $A.P.$
with, $ a=-5 $
$ d=-8-(-5)=-8+5=-3 $
$ I=-230$
Let the number of terms of the $AP$ be $n$
We know that
$ \mathrm{I}=\mathrm{a}+(\mathrm{n}-1) \mathrm{d} $
$ \Rightarrow-230=-5+(\mathrm{n}-1)(-3) $
$ \Rightarrow(\mathrm{n}-1)(-3)=-230+5 $
$ \Rightarrow(\mathrm{n}-1)(-3)=-225 $
$ \Rightarrow n-1=\frac{-225}{-3}=75 $
$ \Rightarrow \mathrm{n}=75+1 $
$ \Rightarrow \mathrm{n}=76$
Again, we know that
$ S_n=\frac{n}{2}(a+l) $
$ \Rightarrow S_{76}=\frac{76}{2}[(-5)+(-230)] $
$ \Rightarrow S_{76}=38(-235) $
$ \Rightarrow S_{76}=-8930$
Hence, the required sum is $-8930 .$
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