Sum of Geometric Sequence

In summary, the conversation discusses finding the sum of a geometric sequence with a given formula. It is first asked to find the sum of all terms, then the sum of alternate terms is asked. The solution involves using partial sums and the formula for the sum of a geometric series.
  • #1
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I'm given the sequence t(n) = 3 (-1)^n (0.5)^n ; n >= 1

It first asks for the sum of the terms t(1) + t(2) + ... + t(99) which is fine, but it follows up by asking the sum of t(1) + t(3) + t(5) + ... + t(99).

Would i be using partial sums to solve this? If so, i don't know how to find the sum of (-0.5)^2n-1 for n = 1 to 99.

Any help would be appreciated, thanks in advance.
 
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  • #2
never mind...

you know, sometimes i wonder about myself...
 
  • #3
Give you a hint...this is a geometric sequence
where 3(-1)^n(0.5)^n = 3(-.5)^n
 
  • #4
as quentinchin said, this is a geometric sequence
t(n) = 3(-1)^n(0.5)^n = 3(-0.5)^n

now,
t(1) + t(3) + t(5) + ... + t(99)
= 3 [(-0.5)^1 + (-0.5)^3 + (-0.5)^5 + ... + (-0.5)^99 ]

now the series inside the brackets [] has first term, a = (-0.5)^1 = -0.5 and common ratio r = (-0.5)^2 = 0.25

therefore,
3 [(-0.5)^1 + (-0.5)^3 + (-0.5)^5 + ... + (-0.5)^99 ]
= 3 [a/(1-r)]
= 3 [-0.5/(1-0.25)]
= 3 [-0.5/0.75]
= -2
 
  • #5
i made a little mistake in my last reply.

t(1) + t(3) + t(5) + ... + t(99)
= 3 [(-0.5)^1 + (-0.5)^3 + (-0.5)^5 + ... + (-0.5)^99 ]

now the series inside the brackets [] has first term, a = (-0.5)^1 = -0.5, common ratio r = (-0.5)^2 = 0.25 and number of terms, n = 50

therefore,
3 [(-0.5)^1 + (-0.5)^3 + (-0.5)^5 + ... + (-0.5)^99 ]
= 3 [a(1 - r^n)/(1-r)]
= 3 [-0.5(1-(-0.5)^50)/(1-0.25)]

but (-0.5)^50 is so close to zero that we can ignore that. therefore,

3 [-0.5{1-(-0.5)^50}/(1-0.25)]
= 3[-0.5(1-0)/(1-0.25)]
= 3 [-0.5/0.75]
= -2
 

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term is multiplied by a constant ratio to get the next term. For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with a common ratio of 3.

What is the formula for finding the sum of a geometric sequence?

The formula for finding the sum of a geometric sequence is Sn = a1(1 - rn)/(1 - r), where Sn is the sum of the first n terms, a1 is the first term, and r is the common ratio.

How do you know if a sequence is geometric?

A sequence is geometric if there is a constant ratio between each term and the previous term. This means that if you divide any term by the previous term, you will always get the same number. Another way to check is to see if the sequence can be written in the form a1, a1r, a1r2, a1r3, ....

Can the sum of a geometric sequence be infinite?

Yes, the sum of an infinite geometric sequence can be infinite if the absolute value of the common ratio (r) is greater than 1. This means that the terms in the sequence will continue to grow larger and larger, resulting in an infinite sum.

What is the difference between a finite and infinite geometric sequence?

A finite geometric sequence has a limited number of terms, while an infinite geometric sequence has an unlimited number of terms. The sum of a finite geometric sequence can be calculated using the formula mentioned earlier, but the sum of an infinite geometric sequence can only be determined if the absolute value of the common ratio is less than 1.

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