Question

There were two ants in the first line, four ants in the second line, six ants in the third line and eight ants in the fourth line, for a total of 20 ants. If this pattern continued, what would be the total number of ants in 15 lines? How can you find out without writing all of the numbers? Can you write an algebraic expression that will allow us to calculate the total number of ants in any row we choose?

Answer 1

The total number of ants in 15 lines is 240 ants.

The expression that allow us to calculate the total number of ants is
`T_a=n^2+n`

The expression that allow us to calculate the total number of ants at any row we choose is
`a=2n`

Explanation:

We see that the total amount of ants is equal line number multiplied by the number of line increased by one. For example:

In two lines we have 2 ants in the first line and 4 in the second one, whose sum is equal 6. If we apply the logic of the above, we should multiply 2 (line number) by 3 (line number +1 ). We can check that 2×3=6.

To calculate the total amount of ants in all the lines, we can find out with the following expression:

`T_a=n * (n+1)`

or

`T_a=n^2+n`

In 15 lines:

`T_a=15^2+15=225+15=240`

In addition, the amount of ants in each line is double the previous one, so we can express the total number of ants in any row that we choose as:

a = 2n