Question

An orchard has 40 rows of trees. Each row is a set number

of trees longer than the one preceding it. If the first row
has 24 trees and the last row has 180 trees, find the
following:
a. How many trees are there in the whole orchard?
b. How many trees are there in the 21st row of the orchard?
ar:

Answer 1

(a) 4,080 trees

(b) 104 trees

Explanation:

If the first row has 24 trees and the fortieth row has 180 trees, how many additional trees are in each succeeding row?

`(40-1)x=180-24`

`39x=156`

`x=4`

That means that each row has 4 more trees than the preceding row. For example, the second row has 28 trees, the third row has 32 trees, and so on. We can write a formula to tell us how many trees are in any given row:

`n=4(r-1)+24`

where
`r` is the row number between 1 and 40.

(a) 24 + 28 + 32 + 36 + 40 + 44 + 48 + 52 + 56 + 60 + 64 + 68 + 72 + 76 + 80 + 84 + 88 + 92 + 96 + 100 + 104 + 108 + 112 + 116 + 120 + 124 + 128 + 132 + 136 + 140 + 144 + 148 + 152 + 156 + 160 + 164 + 168 + 172 + 176 + 180 = 4,080

(b) Using our formula above:

`n=4(21-1)+24`

`n=4(20)+24`

`n=80+24`

`n=104`

Answer 2

a)4080 trees

b)104 trees

Explanation:

1st - 24 trees

last (40th) - 180 trees

If differences between any row (except 40th) and next row are equal then

2nd - 24 trees+X

3rd - (2nd+X) or 24 trees + 2X

coefficient of X is number of row minus 1, therefore 40th row has 24 trees + 39X, and it is equal to 180. We can make an equation from this:

24tr+39X=180tr

39X=156tr

X=4tr

By this we can find how many trees are there in the 21st, 21st row has 24 trees+20X=24 trees+80 trees=104 trees

To find how many trees are there in the whole orchard, we add them all:

1st - 24 trees, 2nd - 24 trees + X, 3rd - 24 trees + 2X,

easily, it is just 24 trees x 40 + 39X x 20 = 4080 trees