Sum of arithmetic series question (1 Viewer)

[kloudsurfer]

Hey,
I have done this question 3 times and i have got a different answer everytime, none of which are right Im sure Ive made a mistake somewhere but I cant figure out where.

Anyway, heres the question:

The sum of 12 terms of an arithmetic series is 186 and the 20th term is 83. Find the sum of 40 terms.


I know that you have to make two equations and solve it simultaneously to find a and d and then find S40, but every time I do it i get it wrong.

Can someone please help me by working this out and show working?

Thanks in advance

by the way, the answer is 3420

 

[SoulSearcher]

Ok, so you are given S₁₂ = 186 and T₂₀ = 83

So 186 = n/2 * [2a+(n-1)d]
n = 12,
186 = 6(2a+11d)
31 = 2a + 11d ... (1)
and
83 = a + 19d ... (2)
[(2)*2] 166 = 2a + 38d ... (3)

Subtract (1) from (3), so you have
135 = 27d
d = 5

Sub d into (2), so that
83 = a + 95
a = -12

So using the sum of arithmetic series formula,
S₄₀ = 40/2 * [2*(-12) + 39*5]
= 20*(-24+195)
= 20*171
= 3420

Hope that helps

 

[watatank]

The sum of 12 terms of an arithmetic series is 186 and the 20th term is 83. Find the sum of 40 terms.

I know that you have to make two equations and solve it simultaneously to find a and d and then find S40, but every time I do it i get it wrong.

Can someone please help me by working this out and show working?

Thanks in advance

by the way, the answer is 3420

S(12) = 186

(n/2) * (2a + 11d) = 186

6(2a + 11d) = 186

2a + 11d = 31


T(20) = 83

a + 19d = 83
2a + 38d = 166

Your two eqns you have to solve simultaneously

(i) 2a + 11d = 31
(ii) 2a + 38d = 166

(ii) - i): 27d = 135

d = 5

sub d into (i)

2a + 11(5) = 31

2a + 55 = 31

2a = -24

a = -12

S(40) = (40/2) * [ 2(-12) + 39(5) ]

= 20(-24 + 195)

= 3420

Hope that helps :wave:

 

[kloudsurfer]

Haha you will not believe what I did! All along ive been writing s(12)=168 not 186!!! I have a weird habit of swapping 6's and 8's!!!!

Thank heaps guys!

 

[bos1234]

they do look kind of similar !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!