F.6 Maths ASGS

2, 6, 12, 20, 30, 42... (original sequence)
4 6 8 10 12 - 1st line of difference between two consecutive terms
2 2 2 2 - 2nd line of difference between two consecutive terms
since the 2nd line of difference is going up by a constant of 2, the general term must involves n^2, the n^2 sequense are as followed:
1, 4, 9, 16, 25, 36... ( n^2 sequense )
compare the n^2 sequence with the original sequence
we can see
n^2 sequense + n = original sequence
for example first term n = 1, 1 + 1 = 2
second term n = 2, 4 + 2 = 6
and so on

so the gerneral term, T(n) = n^2 + n

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Method 1 :
T(1) = 2 = 1*2
T(2) = 6 = 2*3
T(3) = 12 = 3*4
T(4) = 20 = 4*5
T(5) = 30 = 5*6
T(6) = 42 = 6*7
Therefore, T(n) = n(n + 1)

Method 2 :
T(1) = 2
T(2) = 6 = 2 + 4
T(3) = 12 = 2 + 4 + 6
T(4) = 20 = 2 + 4 + 6 + 8
T(5) = 30 = 2 + 4 + 6 + 8 + 10
T(6) = 42 = 2 + 4 + 6 + 8 + 10 + 12
Therefore,
T(n)
= 2 + 4 + 6 + ... + 2n
= (2 + 2n)*n/2
= n(n + 1)