Recursive Formula (Sequences)

Question

The Question is give the recursive formula of each question, and add another 2 terms. 1. 8,14,26,50,... 2. 2,6,-18,-54,... 3. 6,7,7.5,7.75,... 4. 3,10,24,52,...

The problem is that I can't understand the interval in each term for example 8 and 14, the interval is 6 then 14 and 26 is 12, then 24 for 26 and 50. I know that the intervals increases by doubling the last interval. 6*2=12, 12*2=24 and so forth. But I don't know how to get the recursive formula.

Answer 1

Let $a_n$ denotes $n^{\text{th}}$ term of sequence.

You got - \begin{align} &a_2-a_1=6 \times 2^0\\ &a_3-a_2=6 \times 2^1\\ &a_4-a_3=6\times 2^2\\ \end{align}

I hope you see the pattern now

$$a_{n+1}-a_n=6\cdot 2^{n-1}\implies \boxed{\color{blue}{a_{n+1}=a_n+6\cdot 2^{n-1}}}$$

You can proceed similarly in the other two sequences.