One Sequence

Algebra Level 3

Find the 15th term in the following sequence:

$0,\quad 5,\quad 9,\quad 12,\quad 14,\quad 15,\quad 15,\quad 14,\quad ...$

The answer is -21.

4 solutions

Sajid Mamun
Dec 14, 2014

Original Sequence: 0, 5, 9, 12, 14, 15, 15, 14, ...

Find the differences between each term: +5, +4, +3, +2, +1, 0, -1 Find the second differences. -1, -1 ,-1, ... Half the second difference and multiply it by n^2. -1/2n^2

Make a new sequence using -1/2n^2. -0.5, -2, -4.5, -8. -12.5, Subtract the new sequence from the original sequence to find the differences. 0.5, 7, 13.5, 20, 26.5 Find the second differences. 6.5, 6.5, 6.5, ... The sequence 6.5n - 6 must be added to the new sequence to get the original sequence.

Hence the nth term sequence is -0.5n^2 +6.5n-6 .

Substitute 15 as n. -0.5(15^2)+6.5(15)-6 = -21

The answer is -21.

Pranjal Jain
Oct 7, 2014

$a_{n}=\frac{(1-n)(n-12)}{2}$

There is a second possible solution to this which leads to a second possible answer.

$a_n=a_{n-1}+6-n$

with base case $a_0=0$

Trevor Arashiro - 6 years, 8 months ago

I think $a_{1}=0$ . You will get correct answer by this!

Pranjal Jain - 6 years, 8 months ago

Trevor, you will get the same answer. Note that because your first term is $a_0$ , hence the fifteenth term is actually $a_{14}$ , which is equal to -21.

Both of your solutions describe a quadratic form for the answer. If they agree on 3 terms, then they must agree everywhere.

Calvin Lin Staff - 6 years, 8 months ago

Kashif Khan
Nov 10, 2014

set X=5 start with adding X to each term and then set X=x-1

0+5 = 5, 5+4 = 9, 9+ 3= 12, 12+2= 14, 14+ 1= 15, 15+0 = 15, 15+(-1) = 14, 14+ (-2)= 12, . . . . -13+(-8)= -21

Anisur Anis
Oct 14, 2014

-21. after 15 it will gradually decrease , like 1>2>3....

One Sequence

The answer is -21.

4 solutions

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