The number of terms in an ap is even

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the number of terms of an A.P is even; the sum of odd terms is 24 ,of the even term is 30 and last term exceeds the first term by 21/2 , find the number of terms and the series​

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let the total no.of terms be 2n , & first term ,a & common diff. be d

a + (2n-1 ) d = l

& l-a = 10.5

if we take only the odd terms , it will also be a A.p. of n terms with common diff. of 2d

24 = n/2 { 2a + (n-1)2d }

24 = n { a + (n-1 )d } .................................(2)

similarly,

30 = n/2 { 2(a+d) + (n-1)2d }

30 = n/2 { 2a + 2nd }

30 = n { a + nd } ...............................(3)

putting the value of n { a + nd } from eq 3 , in eq 2,

24 = 30 - nd

nd = 6 , putting in eq 1

12 - d = 10.5

d = 1.5 , so n = 6/ 1.5 = 4

putting in eq 3,

30 = 4 ( a + 6 )

a = 1.5

so the no. of terms = 2n = 8

& series ,

1.5 , 3, 4.5 , 6, 7.5, 9, 10.5 . 12

(2n-1 ) d = 10.5 .............. .....

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