PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\]  ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?

Show that \[\lim_{x \to 0} \frac{x}{\left| x \right|}\] does not exist.

The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.

The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.

Find the sum :

\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]

The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S₁ (S₂ + S₃).

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)^th to (2n)^th term is \[\frac{1}{r^n}\].

If a and b are the roots of x² − 3x + p = 0 and c, d are the roots x² − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.

How many terms of the G.P. 3, \[\frac{3}{2}, \frac{3}{4}\] ..... are needed to give the sum \[\frac{3069}{512}\] ?

If S₁, S₂, ..., S_n are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S₁ + S₂ + 2S₃ + 3S₄ + ... (n − 1) S_n = 1ⁿ + 2ⁿ + 3ⁿ + ... + nⁿ.

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.

Let a_n be the nth term of the G.P. of positive numbers.

Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.

Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.

Find the sum of the following serie to infinity:

\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]

Find the sum of the following serie to infinity:

8 +  \[4\sqrt{2}\] + 4 + ... ∞