30. Geometric Progressions Problems Part 2

  1. If \(a\) be the first term, \(b\) be the \(n\text{th}\) term and \(p\) the product of \(n\) terms of a G. P., show that \(p^2 = (ab)^n\).

  2. Show that the ratio of the sums of \(n\) terms of two G.P.’s having the same common ratio is equal to the ratio of the their \(n\text{th}\) terms.

  3. If \(S_1, S_2, S_3\) be the sum of \(n, 2n, 3n\) terms respectively of a G. P., show that \((S_2 - S_1)^2 = S_1(S_3 - S_2)\).

  4. If \(S_n\) denotes the sum of \(n\) terms of a G. P., whose first term is \(a\) and common ratio is \(r\), find \(S_1 + S_2 + ... + S_{2n - 1}\).

  5. The sum of \(n\) terms are is \(a.2^n - b\), find its \(n\text{th}\) term. Are the terms of this series in G. P.?

  6. If \(n\text{th}\) term of a series is \(3.2^n - 4\), find the sum of its 100 terms.

  7. Find the \(n\text{th}\) term and sum to \(n\) terms of the series \(1 + (1 + 2) + (1 + 2 + 2^2) + ...\)

  8. Find \(1 + 3x + 9x^2 + 27x^3 + ... \text{to}~\infty\)

  9. Find \(3 -1 + \frac{1}{3} - \frac{1}{9} ... \text{to}~\infty\)

  10. Find \(\frac{1}{5} + \frac{1}{7} + \frac{1}{5^2} + \frac{1}{7^2} + ... \text{to}~\infty\)

  11. Find the value of

    \[9^{\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + ... \text{to}~\infty}\]
  12. Find

    \[\frac{1}{1 + x^2}\left\{1 + \frac{2x}{1 + x^2} + \left(\frac{2x}{1 + x^2}\right)^2 + ... \text{to}~\infty\right\}\]
  13. Prove that

    \[a^\frac{1}{2}a^\frac{1}{4}a^\frac{1}{8} ... \text{to}~\infty = a.\]
  14. Express \(0.\dot{5}\dot{4}\) as a rational number.

  15. Convert \(0.666666 ... \text{to}~\infty\) into rational number.

  16. The sum of an infinite G. P. whose common ratio is numerically less than \(1\) is \(32\) and the sum of whose first two terms is \(24\). Find the terms of the G. P.

  17. The sum of infinite number of terms of a decreasing G. P. is \(4\) and the sum of the squares of its terms to infinity is \(\frac{16}{3}\), find the G. P.

  18. In an infinite G. P. whose terms are all positive, the common ratio being less than unity, prove that any term \(>, =\) or \(<\) the sum of all the succeeding terms according as the common ratio \(<, =\) or \(> \frac{1}{2}\).

  19. If \(y = 1 + x + x^2 + x^3 + ...\) to \(\infty\) where \(0 < x< 1\), show that \(x = \frac{y - 1}{y}\)

  20. If \(a, b, c\) be three successive terms of a G.P. with common ratio \(r\) and \(a>0\) satisfying the relation \(c > 4b - 3a\) then prove that \(r>3\) or \(r<1\).

  21. If \((1 - k)(1 + 2x + 4x^2 + 8x^3 + 16x^4 + 32x^5) = 1 - k^6\) where \(k\neq 1\) then find \(\frac{k}{x}\).

  22. If \((a^2 + b^2 + c^2)(b^2 + c^2 + d^2) \le (ab + bc + cd)^2\) where \(a, b, c, d\) are non-zero real numbers then shown that they are in G. P.

  23. If \(a_1, a_2, ..., a_n\) are \(n\) non-zero real numbers such that \((a_1^2 + a_2^2 + ... + a_{n- 1}^2)(a_2^2+a_3^2 + ... + a_n^2) \le (a_1a_2 + a_2a_3 + ... + a_{n-1}a_n)^2\) then show that \(a_1, a_2, ..., a_n\) are in G. P.

  24. \(\alpha, \beta\) be the roots of \(x^2 -3x + a = 0\) and \(\gamma, \delta\) the roots of \(x^2 -12x + b = 0\) and the numbers \(\alpha, \beta,\ \gamma, \delta\) form an increasing G. P. then find the values of \(a\) and \(b\).

  25. There are \(4n + 1\) terms in a certain sequence of which the first \(2n + 1\) terms form an A. P. of common difference \(2\) and the last \(2n + 1\) terms are in G. P. of common ratio \(1/2\). If the middle term of both A. P. and G. P. are same then find the mid term of the sequence.

  26. The sum of \(n\) terms of a G. P. is \(3 - \frac{3^{n + 1}}{4^{2n}}\) then find the common ratio.

  27. Three numbers form an increasing G. P. if the middle number is doubled, then the new numbers are in A. P. Find the common ratio of G. P.

  28. If \(f(x) = 2x + 1\) and three unequal numbers \(f(x), f(2x)\) and \(f(4x)\) are in G.P., then find the number of values for \(x\).

  29. Three distinct real numbers \(a, b, c\) are in G. P. such that \(a + b + c = xb\), then show that \(x<-1\) or \(x>3\).

  30. Let \(A\) be the A. M. and \(G\) be the G. M. between two numbers then find the numbers.

  31. If the A. M. and G. M. between two numbers are in the ratio \(m:n\) then find the ratio between the numbers.

  32. If

    \[x = \sum_{n = 0}^{\infty}a^n,~~y = \sum_{n = 0}^{\infty}b^n,~~z = \sum_{n = 0}^{\infty}c^n\]

    where \(a, b, c\) are in A. P., such that \(|a| < 1\), \(|b| < 1\) and \(|c| < 1\) then show that \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in A. P.

  33. Given that \(0 < x < \frac{\pi}{4}\) and \(\frac{\pi}{4} < y < \frac{\pi}{2}\) and

    \[\sum_{k = 0}^{\infty}(-1)^k tan^{2k}x = p,~~\sum_{k = 0}^{\infty}(-1)^k cot^{2k}x = q\]

    then prove that

    \[\sum_{k = 0}^{\infty} tan^{2k}x~~cot^{2k}y\]

    is

    \[\frac{1}{\left\{\frac{1}{p} + \frac{1}{q} - \frac{1}{pq}\right\}}\]
  34. An equilateral triangle is drawn by joining the mid-points of a given equilateral triangle. A third equilateral triangle is drawn inside the second in the same manner and the process is continued indefinitely. If the side of first equilateral triangle is \(3^{1/4}\) inch, then find the sum of areas of all these triangles.

  35. If \(S = exp {(1 + |\cos x| + \cos^2x + |\cos^3x| + \cos^4x ... \text{to}~\infty)\log_e 4}\) satisfies the roots of the equation \(t^2 - 2-t + 64 = 0\) for \(0< x < \pi\) then find the values of \(x\).

  36. If \(S \subset (-\pi, \pi)\) denote the set of values of \(x\) satisfying the equation

    \[8^{1 + |\cos x| + \cos^2 x + |\cos^3 x| + ... \text{to}~\infty} = 4^3\]

    then find the values of \(S\).

  37. If \(|a| < 1\) and \(|b| < 1\), then find the sum of the series \(1 + (1 + a)b + (1 + a + a^2)b^2+ ... \text{to}~\infty\).

  38. If \(0 < x < \pi/2\) and \(exp {(\sin^2 x + \sin^4 x + \sin^4 x + ... \text{to}~\infty)}\) satisfies the roots of the equation \(x^2 - 9x + 8 = 0\), then find the value of \(\cos x/(\cos x + \sin x)\).

  39. If

    \[S_{\lambda} = \sum_{r = 0}^{\infty}\frac{1}{\lambda^r},\]

    then find

    \[\sum_{\lambda = 1}^{n}(\lambda -1)S_{\lambda}.\]
  40. If \(a, b, c\) are in A. P. then prove that \(2^{ax + 1}, 2^{bx + 2}, 2^{cx + 1}\) are in G. P. \(\forall x \neq 0\).

  41. If

    \[\frac{a + be^x}{a - be^x} = \frac{b + ce^x}{b - ce^x} = \frac{c + de^x}{c - de^x}\]

    then prove that \(a,b,c,d\) are in G. P.

  42. If \(x, y, z\) are in G. P. and \(\tan^{-1}x, \tan^{-1}y\) and \(\tan^{-1}z\) are in A. P. then prove that \(x = y =z\) but their common value is not necessarily zero.

  43. If \(a, b, c\) are three unequal numbers such that \(a, b, c\) are in A. P. and \(b - a, c - b, a\) are in G. P. then prove that \(a:b:c\) equals \(1:2:3\).

  44. The sides \(a, b, c\) of a triangle are in G. P. such that \(\log a - \log 2b, \log 2b - \log 3c, \log 3c - \log a\) are in A. P. then prove that \(\triangle ABC\) is an obtuse angled triangle.

  45. If \((a, b), (c, d), (e, f)\) are the vertices of a triangle such that \(a, c, e\) are in G. P. with common ratio \(r\) and \(b, d, f\) are in G. P. with common ratio \(s\) then find the area of the triangle.

  46. If \(\log_t a, a^{t/2}\) and \(\log_b t\) are in G. P. then find the value of \(t\).