Question

conjecture a formula for the terms of the sequence that begins$\mathbf{8}\mathbf{,}\mathbf{14}\mathbf{,}\mathbf{32}\mathbf{,}\mathbf{86}\mathbf{,}\mathbf{248}$ and find the next three terms of your sequence.

Short answer

Hence, the next three numbers of the sequence are:$734,2192,6566$

Step-by-step solution

Step 1

The ${\mathbf{N}}^{\mathbf{th}}$term$a_n=a_1+(n-1)d$

Step 2

$8,14,32,86,248$

This is the given series. At first glance the series solution is not obvious. In these kind of problems, we start by doing simple operations like addition, subtraction and multiplication to see where can get to.

localid="1668422415139" $$\begin{gathered} 8\times 2=16 \\ 16-2=14 \end{gathered}$$This seems a likely possibility

$$\begin{gathered} 14\times 2=28 \\ 28+4=32 \end{gathered}$$May be the sign alternates and its multiplies or powers of 2

Step 3:

$$\begin{gathered} 32\times 2=64 \\ 64-6=580r \\ 64-8=56 \end{gathered}$$However it fails at this point.

localid="1668422418689" $$\begin{gathered} 8\times 3=24 \\ 24-10=14 \end{gathered}$$We try another approach

Step 4:

$$\begin{gathered} 14\times 2=28 \\ 42-20=32 \end{gathered}$$This seems to be working

localid="1668422432782" $$\begin{gathered} 32\times 3=96 \\ 96-10=86 \\ 86\times 3=258 \\ 258-10=248 \end{gathered}$$As we can see, it fits again

Thus the final solution is$x(n+1)=2\times x\left(n\right)-10$