Arithmetic Progression (AP) is a fundamental concept in mathematics, particularly in the field of algebra and sequences. It is a sequence of numbers in which each term after the first one is obtained by adding a constant value to the preceding term. This constant value is known as the “common difference” and is denoted by the letter “d.”
The general form of an AP is given by:
a, a + d, a + 2d, a + 3d, …, a + (n-1)d
where:
- “a” is the first term of the arithmetic progression,
- “d” is the common difference,
- “n” represents the total number of terms in the sequence.
Properties of AP:
- Common Difference: The defining characteristic of an arithmetic progression is that the difference between any two consecutive terms is always the same (constant). For example, if the first term is “a” and the common difference is “d,” then the second term is “a + d,” the third term is “a + 2d,” and so on.
- Nth Term: The “nth term” of an arithmetic progression can be calculated using the formula: Tn = a + (n-1) * d where “Tn” represents the nth term, “a” is the first term, “d” is the common difference, and “n” is the term number.
- Sum of the First “n” Terms: The sum of the first “n” terms of an arithmetic progression can be found using the formulaSn = (n/2) * [2a + (n – 1) * d]where:
- Sn is the sum of the first “n” terms.
- a is the first term of the arithmetic progression.
- d is the common difference between consecutive terms.
- n is the number of terms.
Q. Find the sum of the first 12 terms of the arithmetic progression: 3, 7, 11, 15, …
A) 162
B) 156
C) 300
D) 144
Solution: To find the sum of the first 12 terms (S12) of the AP, we can use the formula for the sum of the first “n” terms:
Sn = (n/2) * [2a + (n – 1) * d]
Given:
a = 3 (the first term)
d = 7 – 3 = 4 (the common difference)
n = 12 (number of terms)
Using the formula:
S12 = (12/2) * [2(3) + (12 – 1) * 4]
S12 = 6 * [6 + 11 * 4]
S12 = 6 * [6 + 44]
S12 = 6 * 50
S12 = 300
So, the sum of the first 12 terms (S12) of the AP is 300.
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