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Calculus Level 3

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.


You can try my other Sequences And Series problems by clicking here : Part II and here : Part I.


The answer is 8.

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3 solutions

Let the number of term be n = 2 m n = 2m , the first term be a a and the difference be d d .

Then, the sum of odd terms:

m [ 2 a + 2 d ( m 1 ) ] 2 = a m + d m 2 d m = 24 . . . ( 1 ) \quad \quad \dfrac {m[2a+2d(m-1)]}{2} = am + dm^2 - dm = 24\quad ...(1)

The sum of even terms:

m [ 2 a + 2 d + 2 d ( m 1 ) ] 2 = a m + d m 2 = 30 . . . ( 2 ) \quad \quad \dfrac {m[2a+2d+2d(m-1)]}{2} = am + dm^2 = 30\quad ...(2)

Equation 2 - Equation 1: d m = 6 \quad \Rightarrow dm = 6

Also the last term exceeds the first by 10.5 10.5 : a + ( 2 m 1 ) d a = 10.5 2 d m d = 10.5 \quad \Rightarrow a + (2m-1)d - a = 10.5 \quad \Rightarrow 2dm - d = 10.5

2 ( 6 ) d = 10.5 d = 1.5 \quad \Rightarrow 2(6) - d = 10.5 \quad \Rightarrow d = 1.5

From d m = 6 m = 4 n = 8 dm = 6 \quad \Rightarrow m = 4 \quad \Rightarrow n = \boxed {8}

Ayush Verma
Nov 16, 2014

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 = n 2 d n 2 d = 30 24 = 6 n d = 12 & ( a n a 1 ) = ( n 1 ) d = 10.5 = 21 2 1 1 n = ( n 1 ) d n d = 21 24 = 1 1 8 n = 8 Let\quad total\quad terms\quad n\quad \& \quad difference\quad is\quad d.\\ \\ { S }_{ even }{ -S }_{ odd }=\sum _{ m=1 }^{ { n }/{ 2 } }{ \left( { a }_{ 2m }-{ a }_{ 2m-1 } \right) } =\sum _{ m=1 }^{ { n }/{ 2 } }{ d= } \cfrac { n }{ 2 } d\\ \\ \cfrac { n }{ 2 } d=30-24=6\quad \Rightarrow nd=12\\ \\ \& \quad \left( { a }_{ n }{ -a }_{ 1 } \right) =\left( n-1 \right) d=10.5=\cfrac { 21 }{ 2 } \\ \\ \Rightarrow 1-\cfrac { 1 }{ n } =\cfrac { \left( n-1 \right) d }{ nd } =\cfrac { 21 }{ 24 } =1-\cfrac { 1 }{ 8 } \\ \\ \Rightarrow n=8

Anand Raj
Dec 6, 2014

A= 1.5 d= 1.5

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