Identify the nth term in the AP sequence: 21, 42, 63, 84, ... Locate the term that corresponds to the value 210.
Which term of an AP, 21, 42, 63, 84, ... is 210?
The 10th term of the given arithmetic progression is 210.
In an arithmetic progression (AP), the nth term a_n can be found using the formula:
a_n = a_1 + (n - 1) * d
where:
- a_n is the nth term,
- a_1 is the first term,
- n is the term number,
- d is the common difference between terms.
Given that the first term a_1 = 21 and the common difference d = 42 - 21 = 63 - 42 = 84 - 63 = 21, we can plug these values into the formula to find the term number for a_n = 210:
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210 = 21 + (n - 1) * 21
Solving for n:
210 = 21 + 21n - 21 210 = 21n n = 210/21 n = 10
So, the 10th term of the given arithmetic progression is 210.
What is an Arithmetic Progression?
An Arithmetic Progression (AP), also known as an arithmetic sequence, is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d).
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The general form of an arithmetic progression can be represented as:
a, a + d, a + 2d, a + 3d, ...
where:
- a is the first term,
- d is the common difference between terms.
For example, in the sequence 2, 5, 8, 11, 14, ..., the first term a is 2, and the common difference d is 3.
Arithmetic progressions are important in various mathematical contexts, including algebra, number theory, and calculus, due to their predictable nature and simple mathematical properties.
