An arithmetic progression has first term a and common difference -1. The sum of the first n terms is equal to the sum of the first 3n terms. Express a in terms of n.

Question

jack1 · Answer

$$a _{n} = a \times k^{n-1}$$
is that arithmetic progression formula?

anonymous · Answer

no its sn = n/2 (U1 +Un)
or S(n)= n/2 (2u1 +(n-1)d)

jack1 · Answer

no, ur right, mine was geometric, sorry
arithmetic is
$$a_n = a + d(n-1) $$
a = first term
d = common difference

jack1 · Answer

so sum is as u described above
@hartnn ...can u help with this plz?

hartnn · Answer

yes, so whats the sum of 1st n terms ? 
n/2 (2a -n+1) right ?
what about sum of 3n terms ?

jack1 · Answer

3n/2 (2a -3n+1)?

hartnn · Answer

yes, 
eddie agree ? you should also try, afterall its your question!

anonymous · Answer

yea i do

hartnn · Answer

good, so just equate them now 
n/2 (2a -n+1) = 3n/2 (2a -3n+1)

notice that n/2 gets cancelled,
after that its just a linear equation, can you simplify ?

anonymous · Answer

so I solve (2a -n+1) = 3 (2a -3n+1)

hartnn · Answer

yes, correct.

hartnn · Answer

but you got all the steps till here ?

anonymous · Answer

no I dont understand how you got there

hartnn · Answer

you have the sum of n terms formula, right ? 
just plug in d=-1, which is given, in that formula what u get ?

hartnn · Answer

and ofcourse, u1 =a

anonymous · Answer

I dont know what you're saying can you type it out

hartnn · Answer

S(n)= n/2 (2u1 +(n-1)d)  is the sum formula
put u1=a and d=-1 in this
what u get ?

anonymous · Answer

S(n) = n/2( 2a + (n-1)-1)
S(n) = n/2( 2a - n +1)

hartnn · Answer

yes, thats what i wrote for sum of n terms, now do the same thing for sum of 3n terms 


S(n)= n/2 (2u1 +(n-1)d)  is the sum formula
put u1=a and d=-1, n=3n in this
what u get ?

anonymous · Answer

S(n) = 3n/2( 2a - 3n +1)

anonymous · Answer

ohhhhhhhhh

hartnn · Answer

yes, and since they are given to be equal, i just equated them.

anonymous · Answer

of so the final solution is 0= 2(2a-4n+1)

hartnn · Answer

how ? 
 (2a -n+1) = 3 (2a -3n+1)
 (2a -n+1) = 6a -9n +3 

continue... ?

anonymous · Answer

i did I made the problem equal to zero
2a -n+1 = 6a -9n +3 
0= 4a -8n +2
0= 2(2a -4n +1)

hartnn · Answer

oh, okk..and u need to isolate a 
2a =4n-1 
a= ... ?

anonymous · Answer

a= 2n -1/2  !!!!!!!

hartnn · Answer

correct! :)
good work.