Sequences and Series, to Find the Sum of First N Terms in an a. P. Arithmetic Mean (a. M.)

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To Find the Sum of First N Terms in an a. P.:

Let be the first term and d be the common difference of an A. P. Let denote the last term, i.e., the term of the A. P. Then,

Let denote the sum of the first terms of the A. P. Then

Reversing the order of terms in the R. H. S. of the above equation, we have

Adding and vertically, we get

containing terms

i.e.,

Also [From]

It is obvious that

Example:

Find the sum of terms.

Solution:

Here

Using the formula , we get

Arithmetic Mean (A. M.):

When three numbers , and are in A. P., then A is called the arithmetic mean of numbers and . We have, or,

Thus, the required A. M. of two numbers and is . Consider the following A. P.:

  • There are five terms between the first term and the last term . These terms are called between and . Consider another A. P.: . In this case there are two arithmetic means, and between and .

  • Generally any number of arithmetic means can be inserted between any two numbers and .

  • Let be arithmetic means between and , then.

is an A. P.

Let be the common difference of this A. P. Clearly it contains terms

term

Now, …………(i)

………….(ii)

………….(n)

These are required arithmetic means between and .

Adding (i), (ii), ..., (n), we get

[Single A. M. between and ]

Example:

Insert five arithmetic means between and .

Solution:

Let and be five arithmetic means between and .

Therefore, are in A. P. with

We have

Hence, the five arithmetic means between and are 1 and .

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