Question

The sum of the first hundred terms of an arithmetic progression with first term 'a' and common difference 'd' is T. The sum of the first 50 odd-numbered terms, i.e. the first, third, fifth,........, ninety-ninth is [(T/2) - 100]. Find the value of a.

Answer 1

it should be 100a in equation 1 ,isn't it?

Answer 2

yes

Answer 3

S(total)=100/2(2a+99d)=T
= 100a+4950d=T ------------- (1)

1,3,5,7,........, note:common diff.=2d
S(odd) = 50/2(2a+98d)=T/2-100
simplify it
= 100a+9800d+200=T-------------(2)

Solve (1) and (2)

Answer 4

Then I got: d= - (4/97)

Answer 5

is it correct ans.?

Answer 6

substitute d in (1)

Answer 7

but there is again two unknown variable.. i.e. T and a.... and again it the result is an equation.

Answer 8

so put d in both and solve both the resulting eq.

Answer 9

u want 3rd equation right?so that u can have 3 equation ,3 unknowns to get a value?
here's how u get 3rd eq:
since, sum of odd numbered term + sum of even numbered term = total sum
so u get even numbered term as T-(T/2-100)=T/2 + 100
with this u can form 3rd equation as 50/28(2(a+d)+98d)=T/2+100
because,1st term here is a+d
now can u solve these 3 equations with 3 unknowns to get a ??

Answer 10

@curiousshubham

Answer 11

* 50/2*(2(a+d)+98d)=T/2+100

Answer 12

Just I checked the answer key and I found the answer as 40 i.e. a = 40.

Answer 13

in the 2nd equation above there should be 4900d instead of 9800d

Answer 14

to get d=4

Answer 15

@hartnn I am really confused.......

Answer 16

@curiousshubham are u sure about the question??is it to find a or to find d only?
and d we easily found out = 4
but we need an extra equation/information to find a,T

Answer 17

@hartnn I am sure about the question and its in my coursebook. I checked the question several times to be sure.

Answer 18

then we both need to seek help from a smarter mathematician :P
maybe @mukushla can help...

Answer 19

@hartnn no problem with ur work...