What Is the Value of 'a' in the Infinite Geometric Series?

In summary, a geometric series with an initial value of a and common ratio r tends to a/(1-r) as the number of terms approaches infinity, assuming -1<r<1. In this case, the given sequence can be simplified to 3(1+a+a^2+...) and is equal to 45/8. Using the formula, we can determine that a/(1-r) = 45/8, which helps us solve for the value of a.
  • #1
lionel messi.
21
0
Sequences and series help...

1. Homework Statement
3+3a+3a^2+...∞ is = to 45/8 where a>0,then a is...?


3. The Attempt at a Solution
since it is a g.p so using
S=(a(rn-1))/(r-1) for r>1
ive all the values except for "n"..can someone help...:/
 
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  • #2


lionel messi. said:
S=(a(rn-1))/(r-1) for r>1
ive all the values except for "n"..can someone help...:/
Wrong formula. This is an infinite geometric series, so use
[tex]a + ar + ar^2 + ar^3 + ... = \frac{a}{1 - r}[/tex]
 
  • #3


thanks..
 
  • #4


n is the number of terms, so in this case there are infinite terms.
As n tends to infinity, what does r^n tend to? (Assuming -1<r<1)
This gets you to the formula that eumyang posted.
 

FAQ: What Is the Value of 'a' in the Infinite Geometric Series?

1) What is a sequence?

A sequence is a collection of numbers, arranged in a specific order, that follows a certain pattern or rule. Each number in a sequence is called a term.

2) What is a series?

A series is the sum of the terms in a sequence. It is represented by the Greek letter sigma (Σ) and the terms are added together using the given pattern or rule.

3) What is the difference between an arithmetic sequence and a geometric sequence?

In an arithmetic sequence, the difference between consecutive terms is constant. For example, 2, 4, 6, 8,... In a geometric sequence, the ratio between consecutive terms is constant. For example, 2, 6, 18, 54,...

4) How do I find the nth term of a sequence?

To find the nth term of a sequence, you can use the formula an = a1 + (n-1)d for an arithmetic sequence, where a1 is the first term and d is the common difference. For a geometric sequence, the formula is an = a1 * r^(n-1), where a1 is the first term and r is the common ratio.

5) How do I determine if a series is convergent or divergent?

A series is convergent if the limit of the sequence of partial sums approaches a finite number. In other words, if the terms of the series get closer and closer to a specific value, the series is convergent. If the limit does not exist, the series is divergent.

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