The number of terms in an arithmetic progression is even; the sum of the odd terms is 24, the sum of the even terms is 30, and the last term exceeds the first by 10.5.
then find the number of terms.
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L e t t o t a l t e r m s n & d i f f e r e n c e i s d . S e v e n − S o d d = m = 1 ∑ n / 2 ( a 2 m − a 2 m − 1 ) = m = 1 ∑ n / 2 d = 2 n d 2 n d = 3 0 − 2 4 = 6 ⇒ n d = 1 2 & ( a n − a 1 ) = ( n − 1 ) d = 1 0 . 5 = 2 2 1 ⇒ 1 − n 1 = n d ( n − 1 ) d = 2 4 2 1 = 1 − 8 1 ⇒ n = 8
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Let the number of term be n = 2 m , the first term be a and the difference be d .
Then, the sum of odd terms:
2 m [ 2 a + 2 d ( m − 1 ) ] = a m + d m 2 − d m = 2 4 . . . ( 1 )
The sum of even terms:
2 m [ 2 a + 2 d + 2 d ( m − 1 ) ] = a m + d m 2 = 3 0 . . . ( 2 )
Equation 2 - Equation 1: ⇒ d m = 6
Also the last term exceeds the first by 1 0 . 5 : ⇒ a + ( 2 m − 1 ) d − a = 1 0 . 5 ⇒ 2 d m − d = 1 0 . 5
⇒ 2 ( 6 ) − d = 1 0 . 5 ⇒ d = 1 . 5
From d m = 6 ⇒ m = 4 ⇒ n = 8