How to Calculate the Sum of the Last n Terms of an Arithmetic Progression (AP)

Introduction to Arithmetic Progressions (AP)

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant called the common difference, denoted as ( d ), to the preceding term. For example, the sequence 2, 5, 8, 11, ... is an AP with the first term ( a_1 2 ) and the common difference ( d 3 ).

Formula for the Sum of the First n Terms of an Arithmetic Progression (AP)

The sum of the first ( n ) terms of an AP, denoted as ( S_n ), is given by the formula:

( S_n frac{n}{2} [a_1 a_n] )

where:

( n ) is the number of terms, ( a_1 ) is the first term, and ( a_n ) is the ( n )-th term.

Another way to express the ( n )-th term in terms of the first term and the common difference is:

( a_n a_1 (n-1)d )

If you need to calculate the sum of the last ( n ) terms of an AP, you will follow the steps outlined below.

Steps to Calculate the Sum of the Last n Terms of an AP

Identify the Last Term (( a_n )): The last term of the AP is the term that ends the sequence of ( n ) terms from the end. Identify the First Term from the End (( a_n' )): The first term from the end is the term that starts the sequence of ( n ) terms from the end. If the common difference is ( d ), the first term from the end can be calculated as: ( a_n' a_n - (n-1)d ) Apply the Sum Formula for the First n Terms from the End: Use the formula for the sum of an arithmetic series where ( a ) is replaced by ( a_n' ) and ( l ) by ( a_n ) to find the sum of the last ( n ) terms. ( S_n frac{n}{2} [a_n' a_n] )

Example

Consider an AP with the last term (( a_n )) as 20, common difference (( d )) as 2, and we want to find the sum of the last 5 terms.

Identify the last term (( a_n )) 20 Calculate the first term from the end (( a_n' )): ( a_n' a_n - (n-1)d ) ( a_n' 20 - (5-1) times 2 ) ( a_n' 20 - 4 times 2 20 - 8 12 ) Calculate the sum (( S_n )) of the last 5 terms: ( S_5 frac{5}{2} [12 20] ) ( S_5 frac{5}{2} [32] ) ( S_5 5 times 16 80 )

Thus, the sum of the last 5 terms is 80.

General Approach for Different Situations

When you need to calculate the sum of the first ( m ) terms, you need to subtract the sum of the first ( m ) terms from the sum of the first ( n ) terms of the entire AP:

( S_m frac{m}{2} [2a_1 (m-1)d] ) ( x S_n - S_m )

This way, you can find the sum of the terms you are interested in, based on the AP's characteristics.

Conclusion: To calculate the sum of the last ( n ) terms of an AP, identify the last term, calculate the first term from the end, and use the appropriate arithmetic series sum formula.