Nth Term of an Arithmetic Sequence


What is the nth term of a arithmetic sequence? Let A(n) be the value of the nth term of the arithmetic sequence.

Let n be the term number in the sequence.

Let A(1) = a be the first term of the sequence.

Then, the function rule for the nth term is A(n) = a + (n - 1)d

Notice that when n = 1,  

A(1) = a + (1 - 1)d

A(1) = a + (0)d

A(1) = a + 0  = a

How to write the function rule for the nth term of an arithmetic sequence


Consider the sequence 2, 5, 8, 11, 14, ...

A(1) = 2 = 2 + 0 × 3

A(2) = 2 + 3 = 2 + 1 × 3

A(3) = 2 + 3 + 3 = 2 + 2 × 3

A(4) = 2 + 3 + 3 + 3 = 2 + 3 × 3
.
.
.

A(n) = 2 + 3 + 3 + 3 + 3 ... + 3 = 2 + (n - 1) × 3

Notice that the factor in blue is one less than its term number shown in red.

The function rule for 2, 5, 8, 11, 14, ... is A(n) = 2 + (n - 1) × 3

Using the function rule of an arithmetic sequence to quickly solve a real life problem


A teenager is trying to raise money for the football club he belongs to. The club came up with an initial amount of 62 dollars. Then, the teenager is able to collect 15 dollars each day from random donors. How much money does he have after 30 days?

Solution

A(n) = 62 + (n - 1) × 15

Notice that when n = 1,

A(1) = 65 + (1 - 1) × 15 = 65 + (0) × 15 = 65 + 0 = 65

Therefore, A(2) is money collected after 1 day, A(3) is money collected after 2 days, and A(31) is money after 30 days.

A(31) = 62 + (31 - 1) × 15

A(31) = 62 + (30) × 15

A(31) = 62 + 450

A(31) = 512

After 30 days, he has 512 dollars.

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