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Maths-

General

Easy

Question

number of terms of an A.P. is even; the sum of the odd terms is 24, and of the even terms is 30, and the last term exceeds the first by 10/2, then the number of terms in the series is

8
4
6
10

The correct answer is: 4

Let the series have $2 n$ terms and the series is $a comma blank a plus d comma blank a plus 2 d comma blank horizontal ellipsis comma blank a plus left parenthesis 2 n minus 1 right parenthesis d$
According to the given conditions, we have
$open square brackets a plus open parentheses a plus 2 d close parentheses plus open parentheses a plus 4 d close parentheses plus horizontal ellipsis plus open parentheses a plus open parentheses 2 n minus 2 close parentheses d close parentheses close square brackets equals 24$
$rightwards double arrow fraction numerator n over denominator 2 end fraction open square brackets 2 a plus open parentheses n minus 1 close parentheses 2 d close square brackets equals 24$
$rightwards double arrow n open square brackets a plus open parentheses n minus 1 close parentheses d close square brackets equals 24$ (1)
Also, $open square brackets open parentheses a plus d close parentheses plus open parentheses a plus 3 d close parentheses plus horizontal ellipsis plus open parentheses a plus open parentheses 2 n minus 1 close parentheses d close parentheses close square brackets equals 30$
$rightwards double arrow fraction numerator n over denominator 2 end fraction open square brackets 2 open parentheses a plus d close parentheses plus open parentheses n minus 1 close parentheses 2 d close square brackets equals 30$
$rightwards double arrow n open square brackets open parentheses a plus d close parentheses plus open parentheses n minus 1 close parentheses d close square brackets equals 30$ (2)
Also, the last term exceeds the first by 21/2. Therefore,
$a plus open parentheses 2 n minus 1 close parentheses d minus a equals 21 divided by 2$
$rightwards double arrow open parentheses 2 n minus 1 close parentheses d equals 21 divided by 2$ (3)
Now, subtracting (1) from (2),
$n d equals 6$ (4)
Dividing (3) by (4), we get
$fraction numerator 2 n minus 1 over denominator n end fraction equals fraction numerator 21 over denominator 12 end fraction$
$rightwards double arrow n equals 4$

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