SOLUTION: Consider the sequence that starts 1, -3, 6, -10, 15 of which 15 is the 5th term. if the 99th term of this sequence is 4950, what is the 100th term?

Algebra ->  Sequences-and-series -> SOLUTION: Consider the sequence that starts 1, -3, 6, -10, 15 of which 15 is the 5th term. if the 99th term of this sequence is 4950, what is the 100th term?      Log On


   

Question 1128108: Consider the sequence that starts 1, -3, 6, -10, 15 of which 15 is the 5th term. if the 99th term of this sequence is 4950, what is the 100th term?
Found 3 solutions by Mtrkcrc, ikleyn, greenestamps:
Answer by Mtrkcrc(8) About Me  (Show Source):
You can put this solution on YOUR website!
-5050
Because 4950+100 is 5050 and turn it to negative the it will be -5050

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
2-nd term = -3 = 1 - 4 = 1+-+2%5E2

3-rd term = 6 = -3 + 9 = -3+%2B+3%5E2

4-th term = -10 = 6 - 16 = 6+-+4%5E2

5-th term = 15 = -10 + 25 = -10+%2B+5%5E2.


The pattern is this recurrent formula  a%5Bn%2B1%5D = a%5Bn%5D+%2B+%28-1%29%5En%2A%28n%2B1%29%5E2.


To find  a%5B100%5D,  take  n+1 = 100 (hence n = 99)  and  a%5B99%5D = 4950  (as it is given).


Then you will get


a%5B100%5D = 4950+%2B+%28-1%29%5E99%2A%2899%2B1%29%5E2 = 4950+%2B+%28-1%29%2A100%5E2 = 4950 - 10000 = -5050.     ANSWER


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My understanding is that not only an answer does matter - the solution (I mean the correct and correctly presented solution) does matter, too.

It is why I wrote this post after the post by @Mtrkcrc.


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Ignoring the signs of the terms for the moment, the sequence is the sequence of triangular numbers. The triangular numbers are the numbers of dots in an array of dots forming triangles of increasing sizes:
The 1st triangular number is 1:
                                            .

The 2nd triangular number 1+2 = 3:
                                            .
                                           . .

The 3rd triangular number is 1+2+3 = 6:
                                            .
                                           . .
                                          . . .

So the formula for the n-th triangular number is the formula for the sum of the positive integers from 1 to n.

In this problem, you are given the 99th term and asked to find the 100th term. Knowing that the given sequence is, if signs are ignored, the sum of the first n positive integers, we can find the 100th term of the sequence (ignoring signs for the moment) by adding 100 to the 99th term.

Then, since the signs are alternating between terms of the actual sequence, we need to change the sign of our answer.

So the simple way to get the 100th term is
(1) 4950+100 = 5050
(2) change the sign to get the answer, -5050