Calculus 2 - Series

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 = ??

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  1. 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

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  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. :)

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  3. 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.

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