SOLUTION: What is the general term of -1,2,-4,8,-16

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Question 1203500: What is the general term of -1,2,-4,8,-16
Found 3 solutions by Theo, MathLover1, Edwin McCravy:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
-1 * -2 = 2
2 * -2 = -4
-4 * -2 = 8
8 * -2 = -16
looks like you have a common multiplier of -2.
a formula would be y = -1 * (-2)^n, i believe.
when n = 0, y = -1
when n = 1, y = -1 * -2 = 2
when n = 3, y = -1 * (-2)^2 = -1 * 4 = 4
when n = 4, y = -1 * (-2)^3 = -1 * -8 = 8
when n = 5, y = -2 * (-2)^4 = -1 * 16 = -16.
etc.....

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:

-1,2,-4,8,-16

since differences between two consecutive terms are not same, sequence is not arithmetic
check if it is geometric sequence
A sequence an where there is a fixed r so that+a%5En%2Fa%5E%28n-1%29=r for all n is called a geometric sequence. The number r is usually called the ratio.
use given terms
-1+and 2
2%2F-1=r+
-2=r+
2+and -4
-4%2F2=r+
-2=r+

The formula for the nth term of this sequence is:

a%5Bn%5D=+a%5B1%5D%2Ar%5E%28n-1%29

since a%5B1%5D=-1+and r=-2, we have

highlight%28a%5Bn%5D=+-1%2A%28-2%29%5E%28n-1%29%29


check the formula:
given that fourth term is a%5B4%5D=8, where n=4
8=+-1%2A%28-2%29%5E%284-1%29
8=+-1%2A%28-2%29%5E3
8=+-1%2A%28-8%29
8=8+ which confirms our formula




Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

General terms for sequences are not unique.  How about this general term:

a%5Bn%5D%22%22=%22%22expr%28-1%2F8%29%2827n%5E4-306n%5E3%2B1197n%5E2-1878n%2B968%29

It's more complicated, but it works:

a%5B1%5D%22%22=%22%22expr%28-1%2F8%29%2827%2A1%5E4-306%2A1%5E3%2B1197%2A1%5E2-1878%2A1%2B968%29%22%22=%22%22expr%28-1%2F8%29%2827%2A1-306%2A1%2B1197%2A1-1878%2A1%2B968%29%7B%7B%7B%22%22=%22%22expr%28-1%2F8%29%2827-306%2B1197-1878%2B968%29%22%22=%22%22expr%28-1%2F8%29%288%29%22%22=%22%22-1

a%5B2%5D%22%22=%22%22expr%28-1%2F8%29%2827%2A2%5E4-306%2A2%5E3%2B1197%2A2%5E2-1878%2A2%2B968%29%22%22=%22%22expr%28-1%2F8%29%2827%2A16-306%2A8%2B1197%2A4-1878%2A2%2B968%29%7B%7B%7B%22%22=%22%22expr%28-1%2F8%29%28432-2448%2B4788-3756%2B968%29%22%22=%22%22expr%28-1%2F8%29%28-16%29%22%22=%22%222

a%5B3%5D%22%22=%22%22expr%28-1%2F8%29%2827%2A3%5E4-306%2A3%5E3%2B1197%2A3%5E2-1878%2A3%2B968%29%22%22=%22%22expr%28-1%2F8%29%2827%2A81-306%2A27%2B1197%2A9-1878%2A3%2B968%29%7B%7B%7B%22%22=%22%22expr%28-1%2F8%29%282187-8262%2B10773-5634%2B968%29%22%22=%22%22expr%28-1%2F8%29%2832%29%22%22=%22%22-4

a%5B4%5D%22%22=%22%22expr%28-1%2F8%29%2827%2A4%5E4-306%2A4%5E3%2B1197%2A4%5E2-1878%2A4%2B968%29%22%22=%22%22expr%28-1%2F8%29%2827%2A256-306%2A64%2B1197%2A16-1878%2A4%2B968%29%7B%7B%7B%22%22=%22%22expr%28-1%2F8%29%286912-19584%2B19152-7512%2B968%29%22%22=%22%22expr%28-1%2F8%29%28-64%29%22%22=%22%228

a%5B5%5D%22%22=%22%22expr%28-1%2F8%29%2827%2A5%5E4-306%2A5%5E3%2B1197%2A5%5E2-1878%2A5%2B968%29%22%22=%22%22expr%28-1%2F8%29%2827%2A625-306%2A125%2B1197%2A25-1878%2A5%2B968%29%7B%7B%7B%22%22=%22%22expr%28-1%2F8%29%2816875-38250%2B29925-9390%2B968%29%22%22=%22%22expr%28-1%2F8%29%28128%29%22%22=%22%22-16

I gave this only as an enrichment example for you.  This is not the general term
your teacher expected you to give.  The other tutors gave you that one.  This is
just to show you that there are more than one general term for any given finite
number of terms of a sequence.

If you calculate the 6th term using this general term, you will find it to be
-211, not 32, which is what you will get as the 6th term using the general term
for a geometric sequence.

Edwin