Sequences and Series

Let \(n\) be an odd integer. If \(\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,}\) for every value of \(\theta ,\) then

Let \({T_r}\) be the \({r^{th}}\) term of an A.P., for \(r=1, 2, 3, ....\) If for some positive integers \(m\), \(n\) we have
\({T_m} = {1 \over n}\) and \({T_n} = {1 \over m},\) then \({T_n} = {1 \over m},\) equals

If \(x > 1,y > 1,z > 1\) are in G.P., then \({1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}\) are in

Let \(p\) and \(q\) be roots of the equation \({x^2} - 2x + A = 0\) and let \(r\) and \(s\) be the roots of the equation \({x^2} - 18x + B = 0.\) If \(p < q < r < s\) are in arithmetic progression, then \(A = \,..........\) and $$B = \,.......

For any odd integer \(n\) \(\ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........\)

The real numbers \({x_1}\), \({x_2}\), \({x_3}\) satisfying the equation \({x^3} - {x^2} + \beta x + \gamma = 0\) are in AP. Find the intervals in which \(\beta \,\,and\,\gamma\) lie.

If \(In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)\) are in A.P., then

For \(0 < \phi < \pi /2,\) if
\(x =\)\(\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty\) then

Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .........

Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and \(\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p\).

If \({S_1}\), \({S_2}\), \({S_3}\),.............,\({S_n}\) are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are \({1 \over 2}\), \({1 \over 3}\), \({1 \over 4}\),...........

If \({\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)\) are in arithmetic progression, determine the value of x.

The number \({\log _2}\,7\) is

If the first and the \((2n-1)\)st terms of an A.P., a G.P. and an H.P. are equal and their \(n\)-th terms are \(a,b\) and \(c\) respectively, then

Sum of the first n terms of the series \({1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............\) is equal to

The sum of the first n terms of the series \({1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........\) is
\(n\,\,{\left( {n + 1} \right)^2}/2,\) when \(n\) is even. When \(n\) is odd, the sum is .............

The solution of the equation \(lo{g_7}\) \(lo{g_5}\) \(\left( {\sqrt {x + 5} + \sqrt x } \right) = 0\) is .............

Find the sum of the series :
$$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms...

If \(a,\,b,\,c\) are in GP., then the equations \(\,\,\alpha {x^2} + 2bx + c = 0\) and \(d{x^2} + 2ex + f = 0\) have a common root if \({d \over a},\,{e \over b},{f \over c}\) are in ________.

If \(n\) is a natural number such that
\(n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}\) and \({p_1},{p_2},\,\,......,\,{p_k}\) are distinct primes, then show that \(In\) \(n \ge k\) $$in...

The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............

If \(a > 0,\,b > 0\) and \(\,c > 0,\) prove that \(\,c > 0,\) prove that \(\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9\)

Find three numbers \(a,b,c\) between \(2\) and \(18\) such that
(i) their sum is \(25\)
(ii) the numbers \(2,\) \(a, b\) are consecutive terms of an A.P. and
(iii) the numbers \(b,c,18\) are consecutive terms of a G.P.

The rational number, which equals the number \(2\overline {357}\) with recurring decimal is

Does there exist a geometric progression containing \(27, 8\) and \(12\) as three of its terms? If it exits, how many such progressions are possible ?

The third term of a geometric progression is 4. The product of the first five terms is

If \(x,\,y\) and \(z\) are \(pth\), \(qth\) and \(rth\) terms respectively of an A.P. and also of a G.P., then \({x^{y - z}}\,{y^{z - x}}\,{z^{x - y}}\) is equal to :

The interior angles of a polygon are in arithmetic progression. The smallest angle is \({120^ \circ }\), and the common difference is \({5^ \circ }\), Find the number of sides of the polygon.

The harmonic mean of two numbers is 4.Their arithmetic mean \(A\) and the geometric mean \(G\) satisfy the relation. \(2A + {G^2} = 27\)