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

$a_5 = 81+80 = 161$