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) (Show Source): Answer by ikleyn(52788) (Show Source): Answer by greenestamps(13200) (Show Source):
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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
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