The First Term of an Arithmetic Progression

In an arithmetic progression, ‘a’ is used for the first term of that A.P. By adding to it the value of common difference, it is possible to find out the further terms of that progression. It means that by using the first term of a progression we can come to know about a lot of properties that are related to a progression. We can find out the sum of the entire sequence. This entity helps us in finding out the terms that are yet not known to us.

The sum of the first ‘n’ terms of AP

There is a certain formula in an arithmetic progression that can be used to find the sum of any number of consecutive terms of an A.P.

The sum of n terms of an A.P. can be given as follows:

S_n = ( n / 2 ) [ 2a + ( n – 1) d ]

Now, let us try to prove this formula true for the first term of an Arithmetic progression.

Finding the sum of only the first term of the given A.P.

S₁ = ( 1 / 2 ) [ 2a + ( 1 – 1) d ]

S₁ = ( 1 / 2 ) [ 2a + ( 0 ) d ]

S₁ = ( 1 / 2 ) [ 2a ]; 0d = 0;

S₁ = a

Hence, it can be said that the formula stands for all the terms of an A.P.

This formula for the sum of an Arithmetic progression can be used only when the nth term for that A.P. is not known to us. However, the case is not as it seems, and we are aware of the nth term of an Arithmetic progression. In that case, the formula for the sum of n terms of an A.P. can be stated in the following way:

S_n = ( n / 2 ) [ a₁ + a_n ]

The formula of n term in AP

It is likely for us to be able to compute any term in an A.P if we are aware of the first term and common difference for that A.P. 

The formula to find any term of the A.P. can be given as follows:

a_n  = a + ( n – 1 ) × d

In this formula,

‘n’ stands for the term that you want to find out for that particular A.P.

‘a’ is the first term of that A.P.

‘d’ is the common difference that has been computed using two consecutive terms of the sequence. 

Let us take n = 1 as an example,

We need to find the first term of an A.P.

Putting the value of n in the formula to find the nth term of an A.P.

a_n = a + ( n – 1 ) × d; n is the term taken;

a₁ = a + ( 1 – 1 ) × d; as n for given A.P. is 1;

a₁ = a + ( 0 ) × d; 0d = 0;

a₁ = a 

Hence, it can be proven that the formula stands true to find out any term of an Arithmetic progression. 

Example: 2, 5, 8, 11, 14…..

Now the first term of the A.P. is 2

The common difference in the sequence has turned out to be 3

Using these values to find the 7th term of this progression:

a_n = a + ( n – 1 ) × d; n stands for the term taken.

a₇ = a + ( 7 – 1 ) × d

a₇ = 2 + ( 6 ) × 3; n – 1 = 6;

a₇ = 2 + 18

a₇ = 20

The first term of the progression

The first term of a progression is just what the name states, the first term of that particular progression. This is the term that gives us an advantage of being able to find out what the other characteristics of that arithmetic progression will be. 

Using the first term of an arithmetic progression it is possible to find the nth term of that A.P. Utilising this we can also find out the common difference for that arithmetic sequence. And, finding out the sum of n terms of that A.P. is also possible. However, we must have the common difference of that sequence to find the sum of the sequence. 

It is possible to make a complete arithmetic sequence even if only the value of the first term and the common difference for that sequence is known to us. An arithmetic sequence using the first term and common difference can be made up in the following way:

a + 0d, a + 1d, a + 2d, a + 3d, ….

As it can be seen, this entire sequence is made up only of the first term and the common difference. The common difference between the term being represented by ‘d’ and ‘a’ is the first term of the sequence. 

Conclusion

It is the first term of an arithmetic progression that initiates a progression. By using this value we can compute any term of that sequence if only the common difference for that progression is known to us. By using the significant values in the formula for the nth term for the A.P. you can find any term that you want. It is also possible to find the sum for n terms of that A.P., and the whole process becomes much more effective if the last term of the A.P. is also known.