we are trying to find the expression to solve for the nth term in the pattern

3, 6, 10, 15, 21

can you help us?

Hi Patrick,

The first thing to do is to look at the differences between successive terms.

  6 -   3 = 3
10 -   6 = 4
15 - 10 = 5
21 - 15 = 6 Hence the next terms are

21 + 7 = 28
28 + 8 = 36
36 + 9 = 45

From the pattern above the sequence can be written

t1 = 3
t2 = 3 + [3]
t3 = 3 + [3 + 4]
t4 = 3 + [3 + 4 + 5]
.
.
.
tn = 3 + [3 + 4 + 5 + ... + (n+1)]

The expression in the square brackets above you can find using the method of Gauss. (There is an easier allpication of this method in an earlier problem.)

There are n - 1 terms in the expression in square brackets so write it as

Sn-1 = 3 + 4 + 5 + ... + (n+1) Now write it backwards

Sn-1 = (n+1) + ... + 5 + 4 + 3
Adding these two expressions (add down) gives

2 Sn-1 = (n+4) + ... + (n+4) + (n+4) + (n+4) = (n-1)(n+4)

Hence

Sn-1 =  (n-1)(n+4)/2
and thus tn = 3 + (n-1)(n+4)/2 = (n+1)(n+2)/2

Cheers,
Denis
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