Question

Find a formula for sn, n≥1, if sn is given by 3/5,4/8,5/11,6/14,7/17,…

Answer 1

s1=(2+n)/[2*(2+n)+(n-2)]
s2=(2+2)/[2*(2+2)+(2-2)]
s3=(2+3)/[2*(2+3)+(3-2)]
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sn=(2+n)/[2*(2+n)+(n-2)]

Answer 2

The formula for sn is $S_n = \frac{2 + n}{2 + 3n}$ where n≥1

The list is given as:

3/5,4/8,5/11,6/14,7/17,…

The above list is a sequence of fractions, where the difference between adjacent numerators is 1, and the difference between adjacent denominator is 3.

The first term of the sequence is:

$S_1 = \frac{3}{5}$

Rewrite as:

$S_1 = \frac{2 + 1}{2 + 3}$

Express 1 and 3 as product

$S_1 = \frac{2 + 1 * 1}{2 + 3 * 1}$

Substitute n for 1

$S_n = \frac{2 + 1 * n}{2 + 3 * n}$

Evaluate the product

$S_n = \frac{2 + n}{2 + 3n}$

Hence, the formula for sn is $S_n = \frac{2 + n}{2 + 3n}$

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