Calculate the first four terms of the following sequence, starting with n = 1 c_n = 3 + 3/2 + 3/3...

Calculate the first four terms of the following sequence, starting with n = 1

c{eq}_n {/eq} = 3 + 3/2 + 3/3 + ... + 3/n.

First, let's rewrite the sequence term a little more succinctly by using summation notation.

{eq}\begin{align*} c_n &= \sum_{k=1}^n \frac3k \\ &= 3 \sum_{k=1}^n \frac1k \end{align*} {/eq}

The first four terms of the sequence are then

{eq}\begin{align*} c_1 = 3 \left( \frac11 \right) &= \boldsymbol{ 3 } \\ c_2 = 3 \left( \frac11 + \frac12 \right) = 3 \left( \frac32 \right) &= \boldsymbol{ \frac92 } \\ c_3 = 3 \left( \frac11 + \frac12 +\frac13 \right) = \frac92 + 1 &= \boldsymbol{ \frac{11}2 } \\ c_4 = 3 \left( \frac11 + \frac12 + \frac13 + \frac14 \right) = \frac{11}2 + \frac34 &= \boldsymbol{ \frac{25}4} \\ \end{align*} {/eq}