We will discuss about some of the properties of Arithmetic Progression which we will frequently use in solving different types of problems on arithmetical progress.
Property I: If a constant quantity is added to or subtracted from each term of an Arithmetic Progression (A. P.), then the resulting terms of the sequence are also in A. P. with same common difference (C.D.).
Proof:
Let {a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), ..............} ........... (i) be an Arithmetic Progression with common difference d.
Again, let k be a fixed constant quantity.
Now k is added to each term of the above A.P. (i)
Then the resulting sequence is a\(_{1}\) + k, a\(_{2}\) + k, a\(_{3}\) + k, a\(_{4}\) + k ..................
Let b\(_{n}\) = a\(_{n}\) + k, n = 1, 2, 3, 4, ............
Then the new sequence is b\(_{1}\), b\(_{2}\), b\(_{3}\), b\(_{4}\), ...............
We have b\(_{n + 1}\)  b\(_{n}\) = (a\(_{n + 1}\) + k)  (a\(_{n}\) + k) = a\(_{n + 1}\)  a\(_{n}\) = d for all n ∈ N, [Since, <a\(_{n}\)> is a sequence with common difference d].
Therefore, the new sequence we get after adding a constant quantity k to each term of the A.P. is also an Arithmetic Progression with common difference d.
To get the clear concept of property I let us follow the below explanation.
Let us assume ‘a’ be the first term and ‘d’ be the common difference of an Arithmetic Progression. Then, the Arithmetic Progression is {a, a + d, a + 2d, a + 3d, a + 4d, ..........}
1. By adding a constant quantity:
If a constant quantity k is added to each term of the Arithmetic Progression {a, a + d, a + 2d, a + 3d, a + 4d, ............} we get,
{a + k, a + d + k, a + 2d + k, a + 3d + k, a + 4d + k, ...........} ............. (i)
First term of the above sequence (i) is (a + k).
Common difference of the above sequence (i) is (a + d + k)  (a + k) = d
Therefore, the terms of the above sequence (i) form an Arithmetic Progression.
Hence, if a constant quantity be added to each term of an Arithmetic Progression, the resulting terms are also in Arithmetic Progression with the same common difference.
2. By subtracting a constant quantity:
If a constant quantity k is subtracted from each term of the Arithmetic Progression {a, a + d, a + 2d, a + 3d, a + 4d, ...........} we get,
{a  k, a + d  k, a + 2d  k, a + 3d  k, a + 4d  k, ...........} ................. (ii)
First term of the above sequence (ii) is (a  k).
Common difference of the above sequence (ii) is (a + d  k)  (a  k) = d
Therefore, the terms of the above sequence (ii) form an Arithmetic Progression.
Hence, if a constant quantity be subtracted from each term of an Arithmetic Progression, the resulting terms are also in Arithmetic Progression with the same common difference.
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Property II: If each term of an Arithmetic Progression is multiplied or divided by a nonzero constant quantity, then the resulting sequence form an Arithmetic Progression.
Proof:
Let us assume {a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), ..............} ........... (i) be an Arithmetic Progression with common difference d.
Again, let k be a fixed nonzero constant quantity.
Let us obtain, b\(_{1}\), b\(_{2}\), b\(_{3}\), b\(_{4}\), ............. be the sequence, after multiplying each term of the given A.P. (i) by k.
b\(_{1}\) = a\(_{1}\)k
b\(_{2}\) = a\(_{2}\)k
b\(_{3}\) = a\(_{3}\)k
b\(_{4}\) = a\(_{4}\)k
..............
..............
b\(_{n}\) = a\(_{n}\)k
............
...........
Now, b\(_{n + 1}\)  b\(_{n}\) = a\(_{n + 1}\)k  a\(_{n}\)k = (a\(_{n + 1}\) – a\(_{n}\))k = dk for all n ∈ N, [Since, <a\(_{n}\)> is a sequence with common difference d]
Therefore, the new sequence we get after multiplying a nonzero constant quantity k to each term of the A. P. is also an Arithmetic Progression with common difference dk.
To get the clear concept of property II let us follow the below explanation.
Let us assume ‘a’ be the first term and ‘d’ be the common difference of an Arithmetic Progression. Then, the Arithmetic Progression is {a, a + d, a + 2d, a + 3d, a + 4d, ............}
1. On multiplying a constant quantity:
If a nonzero constant quantity k (≠ 0) is multiplied by each term of the Arithmetic Progression {a, a + d, a + 2d, a + 3d, a + 4d, ................} we get,
{ak, ak + dk, ak + 2dk, ak + 3dk, .............} ............... (iii)
First term of the above sequence (iii) is ak.
Common difference of the above sequence (iii) is (ak + dk)  ak = dk
Therefore, the terms of the above sequence (iii) form an Arithmetic Progression.
Hence, if a nonzero constant quantity be multiplied by each term of an Arithmetic Progression, the resulting terms are also in Arithmetic Progression.
2. On dividing a constant quantity:
If a nonzero constant quantity k (≠ 0) is divided by each term of the Arithmetic Progression {a, a + d, a + 2d, a + 3d, a + 4d, ...........} we get,
{\(\frac{a}{k}\), \(\frac{a}{k}\) + \(\frac{d}{k}\), \(\frac{a}{k}\) + 2\(\frac{d}{k}\), \(\frac{a}{k}\) + 3\(\frac{d}{k}\), ............} ................. (iv)
First term of the above sequence (iv) is \(\frac{a}{k}\).
Common difference of the above sequence (iv) is (\(\frac{a}{k}\) + \(\frac{d}{k}\))  \(\frac{a}{k}\) = \(\frac{d}{k}\)
Therefore, the terms of the above sequence (iv) form an Arithmetic Progression.
Hence, if a nonzero constant quantity be divided by each term of an Arithmetic Progression, the resulting terms are also in Arithmetic Progression.
Property III:
In an Arithmetic Progression of finite number of terms the sum of any two terms equidistant from the beginning and the end is equal to the sum of the first and last terms.
Proof:
Let us assume ‘a’ be the first term, ‘d’ be the common difference, ‘l’ be the last term and ‘n’ be the number of terms of an A.P. (n is finite).
The second term from the end = l  d
The third term from the end = l  2d
The fourth term from the end = l  3d
The rth term from the end = l  (r  1)d
Again, the rth term from the beginning = a + (r  1)d
Therefore, the sum of the rth terms from the beginning an the end
= a + (r  1)d + l  (r  1)d
= a + rd  d + l  rd + d
= a + l
Hence, the sum of two terms equidistant from the beginning and the end is always same or equal to the sum of the first and last terms.
Property IV:
Three numbers x, y, and z are in Arithmetic Progression if and only if 2y = x + z.
Proof:
Let us assume that, x, y, z be in Arithmetic Progression.
Now, common difference = y  x and again, common difference = z  y
⇒ y  x = z  y
⇒2y = x + z
Conversely, let x, y, z be three numbers such that 2y = x + z. Then we prove that x, y, z are in Arithmetic Progression.
We have, 2y = x + z
⇒ y – x = z – y
⇒ x, y, z are in Arithmetic Progression.
Property V:
A sequence is an Arithmetic Progression if and only if its nth term is a linear expression in n i.e., a\(_{n}\) = A\(_{n}\) + B , where A, B are two constant quantities.
In this case the coefficient of n in an is the common difference (C.D.) of the Arithmetic Progression.
Property VI:
A sequence is an Arithmetic Progression if and only if the sum of its first n terms is of the form An\(^{2}\) + Bn, where A, B are two constant quantities that are independent of n.
In this case the common difference is 2A that is 2 times the coefficient of n\(^{2}\).
Property VII:
A sequence is an Arithmetic Progression if the terms are selected at a regular interval from an Arithmetic Progression.
Property VIII:
If x, y, and z are three consecutive terms of an Arithmetic Progression then 2y = x + z.
`● Arithmetic Progression
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