A lvl H2 Maths: APGP

From http://www.sgforums.com/forums/2297/topics/348847

1) In an AP , 8th term is twice the 4th term and the 20th term is 40. Find the common difference and the sum of the terms from the 8th to the 20th term.

2) The r^th term of an AP is (1+4r). Find in terms of n, the sum of the first n terms.

3) An arithmetic series has first term 1000 and common difference -1.4. Calculate the 1st negative term and the sum of all the positive terms.

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Answer:

1) Given: U₈ = 2U₄, U₂₀ = 40
Apply nth term formula:
a + 7d = 2(a +3d)
a - d = 0 ---(1)
Also, a + 19d = 40 ---(2)
solving simultaneously, d = 2
a = 2

Sum of terms = S₂₀-S₇
= (20/2)(2(2) + (19)(2)) - (7/2)(2(2) + (6)(2))
= (10)(42) - (7/2)(16)
= 364

2) U_r= 1 + 4_r
When r = 1, U₁ = 5
When r = 2, U₂ = 9
When r = 3, U₃ = 13
Thus, common difference = 4

Thus, S_n = (n/2)(2(5) + (n-1)(4))
S_n = (n)(2n + 3)

3) T_n = a + (n-1)d
a = 1000, d = -1.4
Thus, when T_n <> 1001.4
n > 715.286

Hence, the first negative n term is when n = 716

First negative n term = 1000 + (716 - 1)(-1.4) = -1

First negative term is when n = 716
so the last positive term is when n = 715
So, sum of all positive numbers
= (715/2) (2000 + 714 * -1.4)
= 357643