# NCERT Class 11-Math՚S: Chapter – 9 Sequence and Series Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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## 9.1. 3 Important Results on the Sum of Special Sequences

(i) Sum of the first natural numbers:

(iii) Sum of cubes of first natural numbers:

## 9.2 Solved Examples

Question 1:

The first term of an A. P. is , the second term is and the last term is .

Show that the sum of the A. P. is

Let be the common diffrence and be the number of terms of the A. P. Since the first term is and the second term is

Therefore,

Also, the last term is , so

Therefore,

Question 2:

The term of an A. P. is a and term is . Prove that the sum of its terms is

Let A be the first term and D be the common difference of the A. P. It is given that

Subtracting (2) from (1) , we get

Adding (1) and (2) , we get

Now

[ (using … (3) and (4) ]

Question 3:

If there are terms in an A. P. , then prove that the ratio of the sum of odd terms and the sum of even terms is

Let a be the first term and d the common difference of the A. P. Also let be the sum of odd terms of A. P. having terms. Then

Similarly, if denotes the sum of even terms, then

Hence

Question 4:

At the end of each year the value of a certain machine has depreciated by of its value at the beginning of that year. If its initial value was , find the value at the end of years.

After each year the value of the machine is of its value the previous year so at the end of 5 years the machine will depreciate as many times as 5.

Hence, we have to find the 6th term of the G. P. whose first term is and common ratio is . .

Hence, value at the end 5 years

Question 5:

Find the sum of first terms of the A. P. … if it is known that