Arithmetic series — lesson. Mathematics State Board, Class 10.

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In the previous chapter, we explored arithmetic progression. In this chapter, we learn about the arithmetic series.

Arithmetic series are series whose terms are in arithmetic progression.

Consider the \(a, a +d, a +2d, a + 3d +\),... the Arithmetic Progression. The sum of the first '\(n\)' terms of an Arithmetic Progression denoted by \(S_n\) is given by:

\(S_n =\) \(a + (a + d) + (a +2d) + (a + 3d)+... + (a + (n - 1)d)\)...…..….(1)

Let's rewrite the above in reverse order.

\(S_n =\) \( (a + (n - 1)d) + (a + (n - 2)d) +...….+ (a + d) + a\)……...…(2)

Now add (1) and (2) and we get:

\(2S_n =\) \([a + a + (n - 1)d] + [a + d + a + (n - 2)d] +.....+[a + (n - 2)d + (a + d)] + [a + ( n - 1)d + a]\)

\(2S_n =\) \([2a + (n - 1)d] + [2a + (n - 1)d] +....[2a + ( n - 1)d]\) [\(n\) times]

\(2S_n =\) \( n × [2a + (n - 1)d]\)

That is,

S_{n} = \frac{n}{2} (2 a + (n - 1) d)

Therefore, the sum of first '\(n\)' terms of a Arithmetic Progression

S_{n} = \frac{n}{2} (2 a + (n - 1) d)

If the first term \(a\), and the last term '\(l\)' (\(n^t\)\(^h\) term) are given then, we can simplify the expression as follows.

S = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} [a + a + (n - 1) d]

We know that \(l = a + ( n - 1)d\)

Therefore, the equation becomes

S = \frac{n}{2} [a + l]

. We can use this formula to find the sum \(S_n\), when the first term '\(a\)' and the last term '\(l\)' of the \(A.P\) series are the known values.

Important!

Suppose we have the series \(a_1, a_2, a_3,....a_n\)..... then the number of terms in the series is neither negative nor zero. That is the number of terms in the series is always positive.

Let's see an example to understand this concept much better.

Example:

1. Find the sum of the series \(7, 13, 19, 25, 31,..\) upto \(22\) terms.

Solution:

The given series is \(7, 13, 19, 25, 31,..\)

Here, \(a = 7\), common difference \(d = 13 - 7 = 6\). And the total number of term \(n = 22\)

We know the formula to find the sum of the series when \(a, d\) and \(n\) values are known.