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Maths-

General

Easy

Question

If the sum of $m$ terms of an A.P. is the same as the sum of its $n$ terms, then the sum of its $left parenthesis m plus n right parenthesis$ terms is

$m n$
$- m n$
$1 / m n$
0

The correct answer is: 0

Let $a$ be the first term and $d$ be the common difference of the given A.P. Then,
$S subscript m end subscript equals S subscript n end subscript blank rightwards double arrow blank fraction numerator m over denominator 2 end fraction open square brackets 2 a plus open parentheses m minus 1 close parentheses d close square brackets equals fraction numerator n over denominator 2 end fraction left square bracket 2 a plus left parenthesis n minus 1 right parenthesis d right square bracket$
$rightwards double arrow 2 a open parentheses m minus n close parentheses plus left curly bracket m open parentheses m minus 1 close parentheses minus n open parentheses n minus 1 close parentheses right curly bracket d equals 0$
$rightwards double arrow 2 a open parentheses m minus n close parentheses plus open curly brackets open parentheses m to the power of 2 end exponent minus n to the power of 2 end exponent close parentheses minus open parentheses m minus n close parentheses close curly brackets d equals 0$
$rightwards double arrow open parentheses m minus n close parentheses open square brackets 2 a plus open parentheses m plus n minus 1 close parentheses d close square brackets equals 0$
$rightwards double arrow 2 a plus open parentheses m plus n minus 1 close parentheses d equals 0 blank left square bracket because m minus n not equal to 0 right square bracket$ (1)
Now, $S subscript m plus n end subscript equals fraction numerator m plus n over denominator 2 end fraction open square brackets 2 a plus open parentheses m plus n minus 1 close parentheses d close square brackets equals fraction numerator m plus n over denominator 2 end fraction cross times 0 equals 0$ [Using (1)]

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