Question

Find the next three terms in each sequence.

$5,5,6,8,11,15,\mathrm{....}$

Short answer

The next three terms are 20, 26 and 33.

Step-by-step solution

Step 1. Find the common difference of the corresponding terms.

Calculate the common difference between the corresponding terms to find the pattern of the sequence.

$\begin{array}{c}{a}_2-a_1=5-5=0\\ a_3-a_2=6-5=1\\ a_4-a_3=8-6=2\\ a_5-a_4=11-8=3\\ a_6-a_5=15-11=4\end{array}$

Step 2. Find the trend of the given sequence.

Notice that the difference between each corresponding terms are $0,1,2,3\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}4$.

Thus the difference is increasing by 1 in each next terms.

Step 3. Find the next three terms.

Following the pattern the next three terms are calculated as follows.

$$\begin{gathered} a_7-a_6=5 \\ {a}_7=a_6+5 \\ =15+5 \\ {a}_7=20 \\ {a}_8-a_7=6 \\ {a}_8=a_7+6 \\ =20+6 \\ {a}_8=26 \\ {a}_9-a_8=7 \\ {a}_9=a_8+7 \\ =26+7 \\ {a}_9=33 \end{gathered}$$

Therefore, the next three terms of the given sequence are 20, 26 and 33.