Math Sequences Help?

Question

I need help with this word problem. I've learned to do things similar, but nothing quite like this question. Please explain how you arrive at your answer.

How much should be added to each number of the sequence 5,9,21 so that the resulting three numbers form a geometric sequence?

All help is greatly appreciated!

A geometric sequence is one in which you multiply by the previous number by a constant to get the next number.

2,4,8,16,32...
each number is multiplied by 2

Answer 1

I'm trying to post the formula, but I'll have to save little bits as I go so I won't lose the work in case something happens. I already lost it twice!


*****THIS WORKS IF YOU ASSUME THE FIRST EXPONENT OF X IS ZERO!!!*****


It's been a long time, and I always loved math, but geometry wasn't my favorite. Give me an example of a geometric sequence, and I'll probably be able to give you the answer.

Before I saw the word "geometric", I was gonna say add 2, and you would then have prime numbers 7, 11, and 23, but then 13 and 17 are also prime, so that wouldn't have been right.

Got it!

Add -3.

Add negative 3.

5-3=2, 9-3=6, 21-3=18

2 = 2 x 1

6 = 2 x 3

18 = 2 x 9


Restated:

2 = 2 x (3 to the 0 power)

6 = 2 x (3 to the 1 power)

18 = 2 x (3 squared).....(3 sq = 3 to the 2 power)



I'm trying to find a formula for you, but as I said, it's been a long, long time. My husband is helping. He's great at math.



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Refer back to this later:
5 = a + b = a (x to the 0 power) + b

9 = ax + b = a (x to the 1 power) + b

21 = ax² + b = a (x to the 2 power) + b

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If you flip things around and add a to each side, you get this:

Since 5 = a + b, then b = 5-a (subtracted "a" from each side, and switch so the b is on the left side)

Therefore 9 = ax + 5-a

and 4 = ax - a [because you subtracted 5 from each side]

or
4 = a (x-1)

Divide each side by "a" to get
4/a = x-1

Add 1 to each side to get
4/a + 1 = x-1 + 1

Take out the -1 +1 on the right, and you have
4/a + 1 = x or x = 4/a +1
or x = 1 + 4/a


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Okay now,

21 = ax² + 5 - a

Since x = 1 + 4/a, we can change that to:

21 = a (1 + 4/a)² + 5 - a

Then change that to

21 = a (1 + 8/a + 16/a²)

which is the same as

21 = a + 8 + 16/a + 5 - a

change that to read

21 = a ([a+4] /a)² + 5 - a

To make that last one clearer, it's
21 = "a" times ("a plus 4", over "a")², then plus 5 minus "a"



Subtract 5 from each side and you get

16 = ( [a + 4 ] over a )² -a




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ARE YOU STILL WITH ME? Do I need to continue?
++++++++++++++++++++++++

Okay, so we left off with this:

16 = ( [a + 4 ] over a )² -a

16 = a² + 8a + 16 - a²

Take out the a²-a² which equals zero, and subtract 8a from each side.

That leaves 8a = 16




So a = 2

b = 5 - 2 = 3

x = 1 + 4/2 = 1+ 2 = 3

Restated:
a = 2
b = 3
x = 3

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Do you understand it?

My husband wrote it out for me. I get it, but I couldn't have figured out the formula. I got the numbers right, but he had to figure the formula.




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Answer 2

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Answer 3

it's hard, i tried to solve it but.............