An Arithmetic Progression

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The number of terms in an AP is even. The sum of odd and even numbered terms is 24 and 30 respectively. If the last term exceeds the 1st term by 10.5, find the number of terms in the AP.


The answer is 8.

Omkar Pande by Omkar Pande , 22, India

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1 solution

Let the terms be a 1 , a 2 , … , a n a_1, a_2, \ldots , a_n

Then, Note that a 2 − a 1 + a 4 − a 3 + … + a n − a n − 1 = d + d + … + d = n d 2 Since ( a n − a n − 1 = d ) a_2 - a_1 + a_4 - a_3 + \ldots + a_n - a_{n-1} = d + d + \ldots + d = \frac{nd}{2} \text{Since} ( a_n - a_{n-1} = d)

∑ a 2 n − ∑ a 2 n − 1 = n d 2 \sum a_{2n} - \sum a_{2n - 1} = \frac{nd}{2}

n d 2 = 6 \frac{nd}{2} = 6

n d = 12 nd = 12

Also, a n − a 1 = 10.5 a_n - a_1 = 10.5

a + ( n − 1 ) d − a = 10.5 a + (n-1)d - a = 10.5

n d − d = 10.5 nd - d = 10.5

12 − d = 10.5 12 - d = 10.5

d = 1.5 d = 1.5

Therefore n ( 1.5 ) = 12 n(1.5) = 12

n = 8 n = 8

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