If the sum of first m …
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Preeti Dabral 1 year, 8 months ago
Let a be the first term and d be the com m on difference of the given A.P. Then,
Sm = Sn
{tex}\Rightarrow \frac{m}{2}{/tex}{2a + (m - 1) d} = {tex}\frac{n}{2}{/tex}{2a + (n - 1) d}
{tex}\Rightarrow{/tex} 2a(m - n) + {m(m - 1) - n(n - 1)} d = 0
{tex}\Rightarrow{/tex} 2a(m - n) + [(m2 - n2) - (m - n)d = 0
{tex}\Rightarrow{/tex} (m - n){2a + (m + n - 1)d = 0
{tex}\Rightarrow{/tex} 2a + (m + n - 1)d = 0 [{tex}\because{/tex} m - n {tex}\ne{/tex} 0] ...(i)
{tex}\therefore {/tex} Sm+n = {tex}\frac{m+n}{2}{/tex} {2a + (m + n - 1) d} = {tex}\frac{m+n}{2} \times {/tex} 0 = 0 [Using (i)]
Hence proved on basis of above equation.
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