Question 23

If the first and the *n*th term of a G.P. are *a* ad *b*, respectively, and if *P* is the product of *n* terms, prove that *P*^{2} = (*ab*)^{n}.

Answer

The first term of the G.P is *a* and the last term is *b*.

Therefore, the G.P. is *a*, *ar*, *ar*^{2}, *ar*^{3}, … *ar*^{n–1}, where *r* is the common ratio.

*b* = *ar*^{n–1} … (1)

*P* = Product of *n* terms

= (*a*) (*ar*) (*ar*^{2}) … (*ar*^{n–1})

= (*a* × *a* ×…*a*) (*r* × *r*^{2} × …*r*^{n–1})

= *a**n* *r* ^{1 + 2 +…(n–1) }… (2)

Here, 1, 2, …(*n* – 1) is an A.P.

∴1 + 2 + ……….+ (n – 1)

Thus, the given result is proved.

- Q:-
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

- Q:-
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

- Q:-
A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

- Q:- Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
- Q:-
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

- Q:-
How many terms of G.P. 3, 3

^{2}, 3^{3}, … are needed to give the sum 120? - Q:-
Find the sum of all numbers between 200 and 400 which are divisible by 7.

- Q:- Write the following sets in roster form:

(i) A = {x: x is an integer and - 3 < x < 7}.

(ii) B = {x: x is a natural number less than 6}.

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}

(iv) D = {x: x is a prime number which is divisor of 60}.

(v) E = The set of all letters in the word TRIGONOMETRY.

(vi) F = The set of all letters in the word BETTER. - Q:-
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

- Q:-
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

- Q:-
How many terms of G.P. 3, 3

^{2}, 3^{3}, … are needed to give the sum 120? - Q:- Write the following sets in roster form:

(i) A = {x: x is an integer and - 3 < x < 7}.

(ii) B = {x: x is a natural number less than 6}.

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}

(iv) D = {x: x is a prime number which is divisor of 60}.

(v) E = The set of all letters in the word TRIGONOMETRY.

(vi) F = The set of all letters in the word BETTER. - Q:-
What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

- Q:-
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

- Q:-
Find the sum to

*n*terms of the series whose*n*th term is given by*n*(*n*+ 1) (*n*+ 4). - Q:-
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

- Q:-
A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

- Q:-
Find the distance between P (x

_{1}, y_{1}) and Q (x_{2}, y_{2}) when : (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis. - Q:-
If

*f*is a function satisfying f(x +y) = f(x) f(y) for all x,y N such that f(1) = 3and , find the value of

*n*. - Q:-
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?

Nafeesa
2019-10-30 19:06:29

Thank you

Aniska Chatterjee
2019-07-28 08:25:27

Very good.. thanks for the easiest way.

Hitesh Kumar
2019-05-04 15:52:24

Very good

Yash
2019-02-09 12:56:13

Please change the colour of this page And nice explanation

Pranav
2019-02-06 23:30:16

Why n-1/2 is taken instead of n/2

Ravi Ranjan
2018-09-02 00:54:54

Change the colour of the page

Param
2018-06-27 23:56:10

Good

- NCERT Chapter