Commerce (English Medium) इयत्ता ११ - CBSE Question Bank Solutions

Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.

If S denotes the sum of an infinite G.P. S₁ denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively

\[\frac{2S S_1}{S^2 + S_1}\text {  and } \frac{S^2 - S_1}{S^2 + S_1}\]

If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.

The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.

If a, b, c are in G.P., prove that:

a (b² + c²) = c (a² + b²)

If a, b, c are in G.P., prove that:

\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]

If a, b, c are in G.P., prove that:

\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]

If a, b, c are in G.P., prove that:

\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]

If a, b, c are in G.P., prove that:

(a + 2b + 2c) (a − 2b + 2c) = a² + 4c².

If a, b, c, d are in G.P., prove that:

\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]

If a, b, c, d are in G.P., prove that:

 (a + b + c + d)² = (a + b)² + 2 (b + c)² + (c + d)²

If a, b, c, d are in G.P., prove that:

(b + c) (b + d) = (c + a) (c + d)

If a, b, c are in G.P., prove that the following is also in G.P.:

a², b², c²

State the converse and contrapositive of  statement:

If it is hot outside, then you feel thirsty.

State the converse and contrapositive of  statement:

I go to a beach whenever it is a sunny day.