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| Math Central | Quandaries & Queries |
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Question from danii, a student: im trying to solve the nth term for this pattern. 1 3 6 10 15 21 28 any help would be appretiated |
Hi Dani,
Let's write out a table for "n" and the value S(n):
| n | S(n) |
| 1 | 1 |
| 2 | S(1)+2 |
| 3 | S(2)+3 |
| 4 | S(3)+4 |
| 5 | S(4)+5 |
| ... | |
| n | S(n-1)+n |
So you could say that the nth term is just the (n-1)th term plus n. But this may not be what you are seeking. Let's write it another way:
| n | S(n) |
| 1 | 1 |
| 2 | 1+2 |
| 3 | 1+2+3 |
| 4 | 1+2+3+4 |
| 5 | 1+2+3+4+5 |
| ... | |
| n | 1+2+...+(n-1)+n |
So really, S(n) is the sum of the first n natural numbers.
Can you think of a way to express the sum of the first n natural numbers in an easy formula?
Here's a hint:
S(6) = |
(1+2+3+4+5+6) | |
= |
(1+2+3+4+5+6)×2×½ | |
= |
(1+2+3+4+5+6+1+2+3+4+5+6)×½ | |
= |
(1+6+2+5+3+4+4+3+5+2+6+1)×½ | |
= |
((1+6)+(2+5)+(3+4)+(4+3)+(5+2)+(6+1))×½ | |
= |
((6+1)×6)×½ | |
= |
21 (as we expected. | |
Cheers,
Stephen La Rocque.
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