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Formula of the general term of a PA


We already know that in a PAN:

Note that we can write all the terms of a PAN in function of and r:

Therefore, the general term of PAN will be given by the formula:

, no
  • = first term
  • = nth term
  • r = reason
  • no = number of terms

Example 1

Determine the general term of PA (-19, -15, -11,…):

Resolution

The general term of PA (-19, -15, -11,…) é .

Example 2

Determine the 16th term of the PA (3, 9, 15,…):

Resolution

Therefore, the 16th term of PA (3, 9, 15,…) é 93.

Example 3

Interpolate six arithmetic means between -8 and 13:

Resolution

From the statement we have to:

Once reason is found, just interpolate the arithmetic means: (-8, -5, -2, 1, 4, 7, 10, 13).

Example 4

How many multiples of 5 there is between 101 and 999?

Resolution

  • The first multiple of 5 after 101 é 105, therefore = 105;
  • The last multiple of 5 before 999 é 995, therefore = 995;
  • The reason is 5because we are referring to multiples of 5.

Thus, we conclude that there are 179 multiples of 5 in between 101 and 999.

Example 5

Knowing that in a PAN the 2nd term is 9 and the 11th term is 45, write this PAN:

Resolution
Let's write these terms as a function of and r:

We set up a system of equations:

Therefore, the PAN é (5, 9, 13, 17,… ).

Next: Sum of the N Terms of a PA