Question

James has noticed that the number of new tree branches that have grown on a tree in his backyard each year follows the sequence 1,3,5,7,… 1 , 3 , 5 , 7 , … meaning there was 1 1 new branch at 0 0 years, 3 3 new branches at 1 1 year, 5 5 new branches at 2 2 years, and so on.
If f(n)
f
(
n
)
represents the sequence, determine the number of new branches the tree will have at 10
10
years.

Enter the value of f(n)
f
(
n
)
at f(10).
f(10)= [] ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
f
(
10
)
=

[blank]

_
branches

Answer 1

Notice that the pattern is "previous term plus 2".  This is an arithmetic sequence where the difference (d) equals +2

$a_{n}$ = a₁ + d(n - 1)   ; where a₁ is the first term, d is the difference, and n is the term.

f(n) = 1 + 2(n - 1)

f(n) = 1 + 2n - 2

f(n) = 2n - 1

********************************

f(10) = 2(10) - 1

f(10) = 20 - 1

f(10) = 19