General terms for the following infinite sequences

Question

$$S_1 = \{a_n\}_{n=0}^{\infty}=1,\; \frac 32, \;\frac 74, \;\frac {15}8,\cdots$$ $$S_2 = \{b_n\}_{n=0}^{\infty}=5,\; \frac 72,\; \frac {11}4,\;\frac {19}8,\cdots$$

How do I find the general term, i.e. $a_n,b_n$, for each sequence?

Many thanks.

Answer 1

Do you know geometric progressions? Then try "easier" questions:

• Find the pattern in $1,2,4,8,\cdots$
• Find the pattern in $1,3,7,15,\cdots$
• Find the pattern in $5,7, 11, 19,\cdots$

For the second and third ones, try to see how they related to the first one.

Now combine everything together.