PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

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 + ... ∞

Find the sum of the following serie to infinity:

`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`

Find the sum of the following series to infinity:

10 − 9 + 8.1 − 7.29 + ... ∞

Find the sum of the following series to infinity:

`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`

Prove that: (9^1/3 . 9^1/9 . 9^1/27 ... ∞) = 3.

Prove that: (2^1/4 . 4^1/8 . 8^1/16. 16^1/32 ... ∞) = 2.

If S_p denotes the sum of the series 1 + r^p + r^2p + ... to ∞ and s_p the sum of the series 1 − r^p + r^2p − ... to ∞, prove that S_p + s_p = 2 . S_2p.

Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.

Find the rational number whose decimal expansion is `0.4bar23`.