Find the 13th term of the geometric sequence 5, 10, 20, ...
Answers
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Find the 13th term of the given A.P...
The A.P. is 5,10,20,...
So, a = 5 , d = 5 , n= 13...
By the formula...
_______________
an = a + ( n - 1 ) × d
_______________
an = 5 + ( 13 - 1 ) × 5
= 5 + ( 12 × 5 )
= 5 + 60
= 65. ANS....
Hence, the 13th term of the given ap is 65...
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In this tutorial we will mainly be going over geometric sequences and series. We will be going forwards and backwards with this. First we will be given the formula for the nth term and we will be finding specified terms. Then we will turn it around and look at the terms and find the formula for the nth term. We will finish up by looking at geometric series. If you need a review on sequences, feel free to go to Tutorial 54A: Sequences. If you need a review on sequences, feel free to go to Tutorial 54B: Series. I think that you are ready to move ahead.
desk Tutorial
Geometric Sequence
A geometric sequence is a sequence such that each successive term is obtained from the previous term by multiplying by a fixed number called a common ratio.
The sequence 5, 10, 20, 40, 80, .... is an example of a geometric sequence. The pattern is that we are always multiplying by a fixed number of 2 to the previous term to get to the next term.
Be careful that you don't think that every sequence that has a pattern in multiplication is geometric. It is geometric if you are always multiplying by the SAME number each time.
If you need a review on sequences, feel free to go to Tutorial 54A: Sequences.
nth or General Term
of a Geometric Sequence
geometric sequence
where geometric sequence 2 is the first term of the sequence and r is the common ratio.
notebook Example 1: Find the first five terms and the common ratio of the geometric sequence example 1a.
Since a geometric sequence is a sequence, you find the terms exactly the same way that you do a sequence.
example 1e
*n = 4
example 1f
*n = 5
What would be the common ratio for this problem?
If you said 3 you are correct!
Note that you would have to multiply by
What would be the common ratio for this problem?
If you said -1/2 you are correct!
Note that you would have to multiply by -1/2 each time you go from one term to the next: (1)(-1/2) = -1/2, (-1/2)(-1/2) = 1/4, (1/4)(-1/2) = -1/8, and (-1/8)(-1/2)=1/16. It has to be consistent throughout the sequence.
Also note that the base that is being raised to a power is -1/2.
notebook Example 3: Write a formula for the nth term of the geometric sequence 7, 28, 112, 448, .... Do not use a recursive formula.
We will use the nth term formula for a geometric sequence, geometric sequence to help us with this problem.
Basically we need to find two things: the first term of the sequence, geometric sequence 2 and the common ratio, r.
What is geometric sequence 2, the first term?
If you said 7, give yourself a high five. The first term of this sequence is 7.
What is r, the common ratio?
If you said 4, you are right!! Note that you would have to multiply 4 each time you go from one term to the next: (7)(4) = 28, (28)(4) = 112, and (112)(4) = 448. It has to be consistent throughout the sequence.
Putting in 7 for geometric sequence 2 and 4 for r we get:
example 3a
notebook Example 4: Write a formula for the nth term of the geometric sequence 16, - 4, 1, -1/4, .... Do not use a recursive formula.
We will use the nth term formula for a geometric sequence, geometric sequence to help us with this problem.
Basically we need to find two things: the first term of the sequence, geometric sequence 2 and the common ratio, r.
What is geometric sequence 2, the first term?
If you said 16, give yourself a high five. The first term of this sequence is 16.
What is r, the common ratio?
If you said -1/4, you are right!! Note that you would have to multiply -1/4 each time you go from one term to the next: (16)(-1/4) = - 4, (- 4)(-1/4) = 1, and (1)(-1/4) = -1/4. It has to be consistent throughout the sequence.
Putting in 16 for geometric sequence 2 and -1/4 for r we get:
example 4a
notebook Example 5: Find the first term of a geometric sequence with a fifth term
What is geometric sequence 2, the first term?
If you said 3 you are
We will use the formula for the sum of infinite geometric sequence, infinite series 3, to help us with this problem.
Basically we need to find two things: the first term of the sequence and the common ratio.
What is the first term, geometric sequence 2?
If you said 2 you are right!
What is the common ratio, r?
If you said 1/3, give yourself a pat on the back. Note that you would have to multiply 1/3 each time you go from one term to the next: (2)(1/3) = 2/3, (2/3)(1/3) = 2/9, (2/9)(1/3) = 2/27. It has to be consistent throughout the
notebook Example 10: Find the sum of the infinite series example 10a, if possible.
We will use the formula for the sum of infinite geometric sequence, infinite series 3, to help us with this problem.
Basically we need to find two things: the first term of the sequence and the common ratio.
What is the first term, geometric sequence 2?
If you said 1.5 u are right
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