Solving Arithmetic Progressions: Finding the First Term and Sum of the First n Terms

In an arithmetic progression (AP), the sum of the first 10 terms is 50, and the 5th term is three times the 2nd term. This article will guide you through solving the problem to find the first term and the sum of the first 20 terms.

Problem Statement

Given:

The sum of the first 10 terms of an AP is 50. The 5th term is three times the 2nd term.

We need to find the first term and the sum of the first 20 terms.

Solving the Problem Step-by-Step

Step 1: Expressing the Given Information Mathematically

For an arithmetic progression, the sum of the first ( n ) terms (denoted by ( S_n )) is given by:

( S_n frac{n}{2} (2a (n-1)d) )

Where:

( a ) is the first term ( d ) is the common difference ( n ) is the number of terms

The ( n )-th term (denoted by ( T_n )) is given by:

( T_n a (n-1)d )

Given:

( S_{10} 50 )

( T_5 3 times T_2 )

Step 2: Formulating Equations

For the sum of the first 10 terms:

( S_{10} frac{10}{2} (2a 9d) 50 )

( 5 (2a 9d) 50 )

( 2a 9d 10 ) ... 1

For the 5th term being three times the 2nd term:

( a 4d 3 (a d) )

( a 4d 3a 3d )

( 2a - d 0 ) ... 2

Step 3: Solving the System of Equations

From equation 2, we can express ( d ) in terms of ( a ):

( d 2a )

Substitute ( d 2a ) into equation 1:

( 2a 9(2a) 10 )

( 2a 18a 10 )

( 20a 10 )

( a frac{1}{2} )

Now, find ( d ):

( d 2a 2 times frac{1}{2} 1 )

Step 4: Finding the Sum of the First 20 Terms

The sum of the first 20 terms is given by:

( S_{20} frac{20}{2} (2a 19d) )

Plug in the values of ( a ) and ( d ):

( S_{20} 10 (2 times frac{1}{2} 19 times 1) )

( S_{20} 10 (1 19) )

( S_{20} 10 times 20 )

( S_{20} 200 )

Conclusion

The first term ( a ) is ( frac{1}{2} ), and the common difference ( d ) is 1. The sum of the first 20 terms is 200.

Summary of Key Points

The sum of an arithmetic progression can be calculated using the formula ( S_n frac{n}{2} (2a (n-1)d) ). The ( n )-th term of an arithmetic progression is given by ( T_n a (n-1)d ). By solving the system of equations formed from the given conditions, we can find the first term and the common difference.

Related Keywords

arithmetic progression

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant difference ( d ) to the preceding term.

sum of terms

The sum of the first ( n ) terms of an arithmetic progression is given by ( S_n frac{n}{2} (2a (n-1)d) ).

common difference

The common difference ( d ) is the constant amount that each term in the sequence increases or decreases by.