I am so confused on how to do series problems...especially these. How can you tell the pattern and determining the formula for them?
Can someone please help?
26) Write the first five terms of the sequence {an} whose nth term is given.
an = (n + 3)/(2n − 1)
a1 = ?
a2 = ?
a3 = ?
a4 = ?
a5 = ?
27) Find an expression for the nth term of the sequence. (Assume that the pattern continues.)
{3/4, 4/9, 5/16, 6/25, 7/36, ...}
an = ??
Sure! I can help you understand how to solve these series problems.
To determine the pattern and formula for a series, you need to carefully examine the given terms and look for a relationship or pattern between them. Here's how you can approach each problem:
26) Write the first five terms of the sequence {an} whose nth term is given.
The formula for the nth term of the sequence is given as:
an = (n + 3)/(2n - 1)
To find the first five terms, substitute the values of n from 1 to 5 into the formula:
a1 = (1 + 3)/(2(1) - 1) = 4/1 = 4
a2 = (2 + 3)/(2(2) - 1) = 5/3
a3 = (3 + 3)/(2(3) - 1) = 6/5
a4 = (4 + 3)/(2(4) - 1) = 7/7 = 1
a5 = (5 + 3)/(2(5) - 1) = 8/9
Therefore, the first five terms of the sequence are: 4, 5/3, 6/5, 1, 8/9.
27) Find an expression for the nth term of the sequence.
For this problem, we're given the terms {3/4, 4/9, 5/16, 6/25, 7/36, ...}.
Observe the pattern in the sequence: the numerator of each term increases by 1, while the denominator increases by a perfect square (starting from 4).
To find the nth term, we can express the numerator and denominator in terms of n:
Numerator: n + 2
Denominator: (n + 2)^2
Therefore, the expression for the nth term of the sequence is:
an = (n + 2)/(n + 2)^2 = 1/(n + 2)
So, the nth term of the sequence is 1/(n + 2).
I hope this explanation helps you understand how to approach series problems! Feel free to ask if you have any more questions.
an = (n + 3)/(2n − 1)
well, just start plugging in values for n:
a1 = (1+3)/(2*1-1) = 4/1 = 4
a2 = (2+3)/(2*2-1) = 5/3
and so on ...
Look at the numerators:
3,4,5,... = n+2
and the denominators:
4,9,16,... (n+1)^2
So, an = (n+2)/(n+1)^2
Ahh, for the first problem, I wasn't sure if I was supposed to plug in numbers (1-5) or if there was a particular number I was to plug in rather than randomly, but now realizing that it's between a1-a5 with the numbers accordingly.
And for the second problem, I was trying to find a GCF, but it didn't go so well. But I understand it a bit more now. Thank you so much Steve. :)
yeah - sometimes it's possible to overthink this stuff. It just takes practice, like anything else.
Good judgment comes from experience.
Experience comes from bad judgment.