Exercise 42 Page 319

We want to write a recursive formula for the given sequence.

6, 1, 7, 8, 15, 23, \dots

To do so we need to analyze how the consecutive terms are related. Let's find the sum between each pair of consecutive terms.

a_{2} + a_{1} a_{3} + a_{2} a_{4} + a_{3} a_{5} + a_{4} ⋮ = = = = 1 + 6 7 + 1 8 + 7 15 + 8 ⋮ = = = = 781523 ⋮

$\frac{\frac{1}{1}}{\frac{1}{1}} a_{n - 1} + a_{n - 2} = a_{n} \frac{\frac{1}{1}}{\frac{1}{1}}$

We can see above that the sum of the consecutive terms equals the next term. Therefore, to obtain the value of the term in the

n^{th}

position, we need to add two previous terms. With this information and knowing that the first term equals

6

and the second term equals

1,

we can write the recursive formula.

a_{1} = 6, a_{2} = 1 and a_{n} = a_{n - 1} + a_{n - 2}

We will write the next

2

terms of a sequence now.

⋮ a_{6} + a_{5} a_{7} + a_{6} = = ⋮ 23 + 15 38 + 23 = = ⋮ 3861