Question

Find the fifth term in the sequence

Answer 1

Method 1:

The recursive formula states that every next term is three times the previous one, plus two.

So, we have:

• 
  `a_1 = 1`
• 
  `a_2 = 3\cdot 1 + 2 = 3+2 = 5`
• 
  `a_3 = 3\cdot 5 + 2 = 15+2 = 17`
• 
  `a_4 = 3\cdot 17 +2 = 51+2 = 53`
• 
  `a_5 = 3\cdot 53 +2 = 159+2 = 161`

Method 2:

You can write the expression for
`a_5`, and use the recursive formula until you get to the starting point:

`a_5 = 3a_4+2 = 3(3a_3+2)+2 = 9a_3+8 = 9(3a_2+2)+8 = 27a_2+26 = 27(3a_1+2)+26 = 81a_1+80`

Now use the fact that
`a_1 = 1` to get the conclusion

`a_5 = 81+80 = 161`