Nth Term of an Arithmetic Progression

The “nth term” or “general term” of an arithmetic sequence is a formula that allows you to calculate any term in the sequence directly, without having to calculate all the preceding terms. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.

Formulae

The formula for the nth term ($a_n$) of an arithmetic sequence is:

$a_n = a + (n – 1)d$

Where:

$a_n$ = the nth term of the sequence
$a$ = the first term of the sequence
$n$ = the position of the term in the sequence (e.g., 1st, 2nd, 3rd, …)
$d$ = the common difference between consecutive terms

Examples

Example-1: Find the 10th term of the arithmetic sequence: 2, 5, 8, 11, …

Here, $a = 2$ (the first term), $d = 5 – 2 = 3$ (the common difference), and $n = 10$ (we want the 10th term).

Using the formula: $a_{10} = 2 + (10 – 1) \times 3 = 2 + 9 \times 3 = 2 + 27 = 29$.

Therefore, the 10th term is 29.

Example-2: Find the general term of the arithmetic sequence with first term -3 and common difference 4.

Here, $a = -3$ and $d = 4$. We want to find $a_n$ in terms of n.

Using the formula: $a_n = -3 + (n-1) \times 4 = -3 + 4n – 4 = 4n – 7$.

Therefore, the general term is $a_n = 4n – 7$.

Theorem with Proof

Theorem: The general term ($a_n$) of an arithmetic sequence can be determined by the formula $a_n = a + (n – 1)d$.

Proof:

Consider an arithmetic sequence: $a_1, a_2, a_3, a_4, …$

By definition of an arithmetic sequence, the common difference, $d$, is constant:

$d = a_2 – a_1 = a_3 – a_2 = a_4 – a_3 = …$

We can express each term in relation to the first term, $a_1$ (which we denote as $a$):

$a_1 = a$
$a_2 = a_1 + d = a + d$
$a_3 = a_2 + d = (a + d) + d = a + 2d$
$a_4 = a_3 + d = (a + 2d) + d = a + 3d$

Observing the pattern, the nth term, $a_n$, can be written as:

$a_n = a + (n – 1)d$

This completes the proof.

Common mistakes by students

Incorrectly identifying ‘d’: Students sometimes struggle to correctly calculate the common difference. They might subtract terms in the wrong order or make arithmetic errors. Remember to subtract a term from the term that comes directly after it.
Forgetting to subtract 1 from n: The (n-1) part of the formula is frequently overlooked, leading to an incorrect result.
Confusing the formula with other sequences: Students might try to apply this formula to geometric sequences or other types of sequences where it does not apply.

Real Life Application

Arithmetic sequences and the nth term formula have several real-world applications:

Calculating Salary Increases: If your salary increases by a fixed amount each year, your salary forms an arithmetic sequence. The nth term formula helps calculate your salary in a specific year.
Modeling Loan Repayments: If you make equal monthly payments on a loan (ignoring interest for simplicity), the remaining balance forms an arithmetic sequence.
Stacking Objects: The number of objects in rows of a stacked arrangement (e.g., bricks, cans) often follows an arithmetic sequence.

Fun Fact

The ancient Greeks were fascinated by arithmetic sequences. They used them extensively in geometry and number theory. The concept of the “sum of an arithmetic series” (the sum of all the terms in a sequence) was also explored, leading to the famous story of Gauss and his teacher, where young Gauss quickly calculated the sum of the first 100 natural numbers!