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Question

The number of terms in an $$A.P.$$ is even; the sum of the odd terms in it is $$24$$ and that the even terms is $$30$$. If the last term exceeds the first term by $$10\dfrac {1}{2}$$, then the number of terms in the $$A.P.$$ is :


A
4
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B
8
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C
12
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D
16
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Solution

The correct option is B $$8$$
Let $$'a'$$ and $$'d'$$ be the first term and common difference of the series respectively.
Let $$'2n'$$ be the total terms present in the series.
Now, from the given condition we get
$$\displaystyle \frac{n}{2}[2a+(n-1)2d]=24$$ - (for odd terms).............(1)
$$\displaystyle \frac{n}{2}[2a+2d +(n-1)2d]=30$$ - (for even terms)...........(2)
Subtract (1) from (2),
$$nd=6$$.......................(3)
$$a+(2n-1)d-a=10.5$$.............(4)
$$2nd-d=10.5$$
$$d=12-10.5$$.........(from 3)
$$d=1.5$$
$$\displaystyle n= \frac{6}{1.5}=4$$
Total number of terms are $$'2n'$$ i.e. $$8$$

Mathematics

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