My Math Forum Nth term! How?

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 January 31st, 2015, 09:33 AM #1 Newbie   Joined: Nov 2014 From: Cottingham Posts: 8 Thanks: 0 Math Focus: Surds! Nth term! How? The sequence is: 1, 3, 6, 10, 15, 21, 28 So the interval is: 2, 3, 4, 5, 6, 7 And that interval is: 1 Soo, the nth term may be n^2/2 + n/2 But that seems waaaay to complicated, is there an easier way to, say, work out the 100th term (which was the question)
 January 31st, 2015, 09:42 AM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 This sequence of numbers is famous , they are called triangular numbers. $\frac{1}{2}n(n+1)$ You don't have to square anything , one of the numbers , n or (n + 1) will be even so divide it by 2 and multiply the result with the other. Just plug and play. Thanks from v8archie
 January 31st, 2015, 09:47 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2654 Math Focus: Mainly analysis and algebra More explicitly $${n^2 \over 2}+{n \over 2} = \tfrac12(n^2 +n) = \tfrac12n(n+1)$$
January 31st, 2015, 11:26 AM   #4
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Quote:
 Originally Posted by alifeee The sequence is: 1, 3, 6, 10, 15, 21, 28 So the interval is: 2, 3, 4, 5, 6, 7 And that interval is: 1 Soo, the nth term may be n^2/2 + n/2 But that seems waaaay to complicated, is there an easier way to, say, work out the 100th term (which was the question)
\displaystyle \begin{aligned}a_1=1\\ a_2=1+2=3 \\ a_3=1+2+3=6\\ a_4=1+2+3+4=10 \\ .\\.\\. \\ a_n=1+2+3+ ... +n =\dfrac{n(n+1)}{2} \end{aligned} \\\;\\ \\\;\\ a_{100}=\dfrac{100\cdot101}{2}=5050

 January 31st, 2015, 11:49 AM #5 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Or, search the oeis; 1,3,6,10,15,21,28 - OEIS Thanks from CRGreathouse
February 1st, 2015, 09:40 AM   #6
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Quote:
 Originally Posted by alifeee The sequence is: 1, 3, 6, 10, 15, 21, 28 But that seems waaaay to complicated, is there an easier way to, say, work out the 100th term (which was the question)
Yes.

1 = 1 . . . first term

1 + 2 = 3 . . . second term

1 + 2 + 3 = 6 . . . third term

1 + 2 + 3 + 4 = 10 . . . fourth term

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Sum = 1 + 2 + 3 + . . . + 98 + 99 + 100 . . . hundredth term
Sum = 100 +99 + 98 + ... + 3 + 2 + 1 . . . . . hundredth term

Twice the sum = 101 + 101 + 101 + ... + 101 + 101 + 101

There are 100 columns.

2*Sum = 100*101

Sum = (100*101)/2

You can finish it.

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### 1,3,6,10.......,5050 the nth term of it is?

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