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If the sum of first m term of an a.p is the same as the sum it's first n term show that the sum of its first m+n term is zero

Posted by Ashutosh Singh 2 years, 1 month ago

CBSE > Class 10 > Mathematics

1 answers

Preeti Dabral 2 years, 1 month ago

Let a be the first term and d be the com m on difference of the given A.P. Then,
S_m= S_n
$\Rightarrow \frac{m}{2}$ {2a + (m - 1) d} = $\frac{n}{2}$ {2a + (n - 1) d}
$\Rightarrow$ 2a(m - n) + {m(m - 1) - n(n - 1)} d = 0
$\Rightarrow$ 2a(m - n) + [(m²- n²) - (m - n)d = 0
$\Rightarrow$ (m - n){2a + (m + n - 1)d = 0
$\Rightarrow$ 2a + (m + n - 1)d = 0 [ $\because$ m - n $\ne$ 0] ...(i)
$\therefore$ S_m+n= $\frac{m+n}{2}$ {2a + (m + n - 1) d} = $\frac{m+n}{2} \times$ 0 = 0 [Using (i)]
Hence proved on basis of above equation.

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CBSE > Class 10 > Mathematics