Sum of n terms of an Arithmetic Progression
Properties of Arithmetic Progression
1. If a, b, c are in A.P then 2b = a + c.
2. If a, b, c are in A.P and k ≠ 0, then the following are also in A.P
(i). a + k, b + k, c + k
(ii). a – k, b – k, c – k.
(iii). ak, bk, ck
(iv). a/k, b/k, c/k.
Sequence: A set of numbers arranged in a definite order according to some definite rule (or rules) is called a 'sequence'. Each number of the set is called a 'term' of the sequence.
Finite and Infinite sequence: A sequence is called finite or infinite according to the number of terms in it is finite or infinite.
General term of sequence: Let us consider the sequence of "cubes" of natural numbers: 1, 8, 27, 64, . . . . The different terms of a sequence are usually denoted by t1, t2, t3, . . . etc. Here, the subscript (always a natural number) denotes the position of the term in the sequence. Thus, in the above sequence t1 = 1, t2 = 8, t3 = 27 . . . . . etc. Hence, First term = t1 = 1, Second term = t2 = 8, Third term = t3 = 27, . . . etc. In general, nth term = tn, which is called "general term" of the sequence.
Often, it is possible to express the rule which generates the various terms of the sequence in terms of algebraic formula. In the above sequence 1, 8, 27, 64, 125 . . ., nth term = tn = n3. Thus, the rule for the above sequence is n3, where n is any natural number.
Series: If a1, a2, . . ., an is a sequence of numbers, then the expression a1 + a2 + . . . + an + . . . is called series associated with the given sequence.
A "series" may be finite or infinite. It is common to represent a series compactly using the Σ (sigma) symbol indicating a summation as:
- ai = a1 + a2 + a3 + . . . + an for a finite series.
- ai = a1 + a2 + a3 + . . . + an + . . . for an infinite series.
Note that 'i' takes the values from '1' to the number indicated at the top of the sigma symbol.
Arithmetic progression: An arithmetic progression is a sequence in which each term, except the first term is obtained by adding a fixed number (positive or negative or zero) to the term immediately preceeding it. This fixed number is the difference of two successive terms, for this reason it is called as the common difference usually denoted by d.
Quantities are said to be in A.P when they increase or decrease by a common difference. The common difference is formed by subtracting any term of the sequence from that which follows it.
|Thus, If t1, t2, t3, . . . . . , tn are the terms in an A.P and the common difference is d then
|t2 = t1 + d d = t2 – t1 or t3 = t2 + d d = t3 – t2 or t4 = t3 + d d = t4 – t3
|t5 = t4 + d d = t5 – t4 or t6 = t5 + d d = t6 – t5 or t7 = t6 + d d = t7 – t6
|- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - tn = tn - 1 + d d = tn – tn - 1
|In above, If the first term t1 = a, then t2 = t1 + d = a + d = a + (2 – 1)d
|t3 = t2 + d = [a + d] + d = a + 2d = a + (3 – 1)d
|t4 = t3 + d = [a + 2d] + d = a + 3d = a + (4 – 1)d
|t5 = t4 + d = [a + 3d] + d = a + 4d = a + (5 – 1)d
|- - - - - - - - - - tn = tn – 1 + d = a + (n – 1)d, which is called "general term of A.P".
|By substituting n = 1, 2, 3, . . . we get a, a + d, a + 2d, a + 3d, a + 4d, . . . . . . . . represents an arithmetic progression where "a" is the first term and "d" is the common difference. This is called the "general form of an A.P" and the common difference is given by "tn – tn – 1".
Arithmetic mean: When three quantities are in A.P. the middle one is said to be the arithmetic mean of the other two. Thus, if a, b, c are three quantities in A.P then b is called the arithmetic mean of a, c and it is given by b =. Hence, arithmetic mean of two numbers is half of their sum.