Finding First Term in Geometric Progression with Given Terms?

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In summary, the first term (a) in a geometric progression can be found using the formula ar^{n-1} where r is the common ratio and n is the term number. If two terms are known, the ratio can be found by taking the (j-i)th root of the two terms divided. Once the ratio is known, the first term can be solved for using either of the original equations. In the example given, the ratio was found to be 256 and the first term was either 4 or -4.
  • #1
rush
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Hi there,
Can anybody help please.

How can i find the first term in a geometric progression if i know that the 4th term = 256 and the 8th term = 65536?
 
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  • #2
Those are both powers of 2.
 
  • #3
Sorry, perhaps that wasn't a good example, in general how do i find the 1st term (say a) in a geometric progression if i know 2 other other terms (say b and c)
 
  • #4
Use logarithmation.If 65536 is the 8-th term,then the first is [itex] \log_{8} 65536 [/itex]...(the first is just the ratio.If it's not,then u need two terms...)

Daniel.
 
  • #5
It turns out the answer is -4

There must be a formula to work this out without using logarithmation
 
  • #6
Actually, it's [itex]\pm 4[/itex]. See why?
 
  • #7
What is the definition of the n'th term of a GP with initial term a and ratio r?

ar^{n-1}, right?


so given two terms you've two unknowns and fortunately you can solve for them, though at some point you will need to take some roots.
 
  • #8
The general term of a geometric sequence is arn-1 where a is the first value (n= 1) and r is the "common ratio". If you know "the i th term is x" then you know ari-1= x. If you know "the jth term is y" then you know arj-1= y. Divide the second equation by the first and the "a"s cancel: rj-1/ri-1= rj-i= y/x. Now it's not really necessary to use logarithms to find r- just use the "j-i" root. Once you know r, you can solve either of the original equations for a.

In the example you started with, 4th term = 256 and the 8th term = 65536,
we know ar3= 256 and ar7= 65536 so, dividing the second equation by the first, r7/r3= r4= 65536/256= 256.
Since r= 256, a(2563)= 256 = 1/2562= 1/65536.
 
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  • #9
Halls,[itex] r^{4}=256 [/itex] has 4 solutions...[itex] \left\{\pm 4,\pm 4i\right\} [/itex]...

Daniel.
 
  • #10
Since r= 256, a(2563)= 256 = 1/2562= 1/65536.

you mean [itex]r^4 = 256 \Longrightarrow r = \pm 4[/itex] (over [itex]\mathbb{R}[/itex]) so

[tex]a(\pm 4)^3 = 256 \Longrightarrow a =\biggr \{ \begin{array}{cc} 4 & \mbox{if} \ r=4 \\ -4 & \mbox{if} \ r = -4\end{array}[/tex]

or just [itex]a = r = \pm 4[/itex].
 
  • #11
Data said:
you mean [itex]r^4 = 256 \Longrightarrow r = \pm 4[/itex] (over [itex]\mathbb{R}[/itex]) so

[tex]a(\pm 4)^3 = 256 \Longrightarrow a =\biggr \{ \begin{array}{cc} 4 & \mbox{if} \ r=4 \\ -4 & \mbox{if} \ r = -4\end{array}[/tex]

or just [itex]a = r = \pm 4[/itex].

Oops. My math is fine- but my arithmetic is terrible!
 

FAQ: Finding First Term in Geometric Progression with Given Terms?

What is a geometric progression?

A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio.

How is a geometric progression different from an arithmetic progression?

In an arithmetic progression, the difference between consecutive terms is constant, while in a geometric progression, the ratio between consecutive terms is constant.

What is the formula for finding the nth term in a geometric progression?

The formula for finding the nth term in a geometric progression is an = a1(r)n-1, where an is the nth term, a1 is the first term, and r is the common ratio.

How can a geometric progression be used in real life?

Geometric progressions can be used to model population growth, compound interest, and the growth of bacteria and viruses. They can also be used in financial planning and analyzing stock market trends.

Are there any limitations to using geometric progressions?

Yes, geometric progressions are only appropriate for situations where the common ratio remains constant. In real life, this is often not the case, so geometric progressions may not accurately represent the situation. Additionally, geometric progressions can only be used for situations with positive values, as negative values would result in an alternating sequence.

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