Question

What formula would I use to find the tallest fence pale?

A fence is built behind the oak tree. The fence is made from pales and is sloping upwards. There are 30 pales in the sloping fence. The shortest fence pale is 1.4 metres high. Each fence pale is 40 millimetres taller than the previous pale.

Find the height of the tallest fence pale

Answer 1

`a_n=1.4+0.04(n-1)` gives the last, or tallest, pale as being `2.56` meters.

Explanation:

First, let's make sure we're being consistent with our units. Everything should be in the form of one unit... let's choose meters. Converting `40` millimeters to meters yields `0.04` meters.

This problem deals with an arithmetic sequence: every pale of the `30` pales is `0.04` meters taller than the previous pale; this can be compared to a sequence of `30` terms in which every term is `0.04` more than the last (IE, the difference between each term is `0.04`) .

Our first pale, or first term in our sequence, is `1.4` meters.

We can think of our sequence as being represented by

`a_n=1.4+0.04(n-1)`, derived from the general formula for an arithmetic sequence

`a_n=a_1+d(n-1)` where `a_1` is the first term in the sequence and `d` is the difference between each term.

where `n` represents the `nth` fence pale. We want the `30th` fence pale. Since every pale is taller than the last pale, the tallest pale must also be the last (or `30th`) pale. Thus, we want `a_30`.

`a_30=1.4+0.04(30-1)=1+0.04(29)=2.56` meters