Arithmetic sequence – formula

The nth term of an arithmetic sequence is given by Tn = a +( n-1) d π

The first term is denoted by a. π

The common difference between terms is denoted by d. π

Common difference = any term – previous term π

If you have a and d, you can find any term of the sequence.

This formula is all about the location of terms.

Example: Which term has a value of 126 in the arithmetic sequence 4, 6, 8, … ? Theoretically, you could write out all of the terms until you get to the number 126. The faster and more efficient approach is the Tn

formula. Here, the first term is 4 and the difference is 2. Hence, a =4and d =2 respectively. Finally, we want to find the term that has a value of 126. Hence, Tn = 126. Solution:

Tn =a+( n –1)d

126=4+( n – 1)(2)

126=4+2 n –2

⇒

124= 2n n =62

This means that the 62nd term has the value 126.

What is the difference between a sequence and a series? This is a common question that trips students up and I tend to ask it first when revising the topic. The only difference between a sequence and a series is that the comma signs are replaced with plus signs.

For example, 7, 14, 21, 28, 35, …, is an arithmetic sequence whereas 7 + 14 + 21 + 28 + 35 + …, is an arithmetic series.

We use the symbol Sn

to denote the sum of the first n terms of a series. For example, S12 is the sum of the first twelve terms of the sequence. A series is the sum of the individual terms of a sequence.