4.2 Arithmetic Progressions

The first three terms in an arithmetic sequence are 

Find the possible values of .

An arithmetic sequence has first term  and common difference , where .  The fourth term in the sequence is 4.

Find the value of , giving your answer as an exact value.

The third term of an arithmetic series is 32.  The eleventh term is 0.  The sum of the first  terms is -44.

Find the value of .

The sum of the first twelve terms in an arithmetic series is 654.  The sum of the first twenty terms in the same series is 530.  Find the 21^st term.

Prove that the sum of the first  odd numbers is a square number for any value of .

The first three terms of an arithmetic progression are , and ,  where  is an integer.  The th term is 1008.

In terms of, find an expression for the number of terms in this progression.

In addition to being an integer, what other two conditions must  satisfy for this to be a valid arithmetic progression?

Show that the sum of the first  terms of the progression is 

The seventh term of an arithmetic series is , where  is a constant, and the sum of the first nine terms of the series is .

In terms of , find expressions for the first term and common difference of the series.

Given that the nineteenth term of the sequence is 57, find the sum of the first 25 terms of the series.

The first three terms in an arithmetic progression are    and .  The th term of the progression is denoted by .

Find an expression for in terms of  and .

For a particular value of ,   and the sum of the first terms of the progression is .

Find the value of .

The th term of an arithmetic progression is given by 

Calculate

The sum of the fifteenth through twenty-fifth terms of the progression.

The sum of the first 25 terms of the progression whose th term is given by ,  where is defined as above.

The first three terms in an arithmetic sequence are 

Find the possible values of .

An arithmetic sequence has first term  and common difference , where .  The fifth term of the sequence is 85.

Find:

the value of 

The third term of an arithmetic series is 2.  The twelfth term is 65.  The sum of the first  terms is 390.

Show that  .

Hence find the value of .

The sum of the first ten terms in an arithmetic series is 40.  The sum of the first twenty terms in the same series is 280.  Find the first term, , and the common difference, , of the series.

The first three terms of an arithmetic progression are   and . The th term is 36.

      Show that  

Hence show that the sum of the first   terms is  

Given that the sum of the first terms is 180, find the value of .

The fifth term of an arithmetic series is , where  is a constant, and the sum of the first eight terms of the series is .

Show that the first term, , of the series is .

Find an expression for the common difference, , of the series in terms of .

Given that the ninth term of the sequence is 14, calculate:

the value of 

The sum of the first 30 terms of the series.

The th term of an arithmetic progression is given by.

Calculate

The sum of the first ten terms of the progression.

The sum of the eleventh through fifteenth terms of the progression.

Calculate the sum of all the odd numbers between 0 and 150,

Below is an arithmetic progression

where  is an integer and a positive factor of 360.

In terms of , find an expression for the number of terms in the progression.

Show that the sum of the terms in the progression is 

The first three terms of an arithmetic sequence are  and .  In terms of q , find the 100th term of the sequence.  Give your answer in simplest form.

The first three terms in an arithmetic sequence are 

Given that the first three terms in the sequence are all positive, find the fortieth term in the sequence.

The th term of an arithmetic progression is given by Given that the sum of the first terms is equal to ,  find the value of .

The ^th term of an arithmetic series is 0.  The sum of the first  terms is also 0.

Find the value of in terms of  , giving clear reasons for your answer.

The sum of the first 20 terms in an arithmetic series is 290.  The sum of the first 24 terms in the same series is -180.  In general, the sum of the first  terms is .  Find the greatest value attained by  for any

The sum of the first 24 terms in an arithmetic series is nine times the sum of the first two terms in the series.

Find the sum of the first 90 terms in the series.

An arithmetic progression has first term  and common difference . 

Given that

the sum of the seventh through twelfth terms is ,

the sum of the seventh through sixteenth terms is ,  and

the sum of the first six terms is ,

find the values of  and .