Formula of common difference of an A.P
So, the formula for finding the common difference is,
Formula of common difference if the sequence is given,
Let the sequence be, a1, a2, a3,……,an-1, an
Now, the common difference in the sequence is
d = a2 – a1 = a3 – a2 = …… = an – an-1
Hence,
d = an-an-1
where,
an is the nth term and
an-1 is its preceding term.
Formula of common difference when the nth term and the first term of the sequence are given,
Let “a” be the first term and “d” be the common difference in the arithmetic sequence. Now, the nth term of the sequence is,
an = a + (n-1) d
By subtracting the first term from the nth term, we get
⇒ an – a1 = a + (n–1) d – a
⇒ an – a1 = (n–1)d
⇒ d = (an – a1)/(n – 1)
Hence,
d = (an -a1)/(n-1)
where,
an is the nth term and
a1 is the first term.
Formula of common difference when the sum of n terms and the first term of the sequence are given,
Let “a” be the first term and “d” be the common difference in the arithmetic sequence.
The sum of n terms of an arithmetic sequence is,
Sn = (n/2)[2a + (n-1) d]
⇒ Sn × (2/n) = 2a + (n-1)d
⇒ (Sn × 2/n) – 2a = (n-1)d
⇒ d = [(Sn × 2/n) – 2a]/(n-1)
Hence,
d = [(Sn×2/n) – 2a1)]/(n-1)
where,
Sn is the sum of n terms and
a1 is the first term.
How to Find the Common Difference of an Arithmetic Progression?
How to find the common difference of an Arithmetic Progression? A sequence is also known as a progression and is defined as the successive arrangement of numbers in order while following some specific rules. Depending upon the set of rules followed by a sequence, it is classified into various kinds, such as an arithmetic sequence, geometric sequence, harmonic sequence, and Fibonacci sequence. An arithmetic sequence or progression is a sequence of numbers when the difference between any two successive numbers is the same. For instance, 5, 10, 15, 20, 25, 30,… is an arithmetic sequence, where the common difference between any two successive terms is 5.
General Form of an AP:
AP = a, a + d, a + 2d, a + 3d,……., a + (n-1) d
Where,
“a” is the first term of the progression, and
“d” is a common difference.