If the sum of first n even natural numbers is equal to m times the sum of first n is odd natural numbers then m is equal to ______.

The sum to infinity of \(\frac{1}{3}+\frac{3}{9}+\frac{5}{27}+\frac{7}{81}+...+\infty\) is ______.

The sum of the square of (n - 1) natural numbers is ______.

The seen to infinity of the series \(1+2.\frac{1}{2}+3.\frac{1}{2^2}+4.\frac{1}{2^3}+...+\infty\) ______.

The sum of infinity of the G.P. a, ar, ar², ar³, ...... \(\infty\) is ______.

Find two numbers whose A.M. is 34 and G.M. is 16

A square is drawn by joining the mid points of the sides of a square. A third square is drawn inside the second square in the same way and the process is continued indefinitely. If the side of the square is 10 cm. Find the sum of the areas of all the squares so formed

If third and fourth terms in the expansion of (a + b)ⁿ are in the ratio as the fourth and fifth terms in (a + b)ⁿ⁺³, find the value of n.

Using Geometric series, find the rational number whose decimal expansion is 0.142.

The sum of an infinite G.P. is 57 and the sum of their cubes is 9747, find the G.P.

The (m+n)^th and (m -n)th terms of G.P are P and q respectively. Show that the m^th and n^th terms are \(\sqrt { pq } \) and \(p\left( \frac { p }{ q } \right) ^{ \frac { m }{ 2n } }\) respectively

Determine the number of terms in the A.P. 4, 8, 12, 288. Also find its 18th term from the end.

The fourth term of a C.P. is 4. Find the product of its first seven terms.

Find the sum to infinity of the series (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + .......

If S1, S2, S3, .... SP denote the sums of infinite geometric series whose first terms are 1, 2, 3, .... P respectively and whose common rations are \(\frac{1}{2},\frac{1}{3},\frac{1}{4},....,\frac{1}{P-1}\) respectively Prove that \(S_1+S_2+S_3+..+S_P=\frac{P(P+3)}{2}\)

The 2nd, 31st and last term of an AP are 7\(\frac { 2 }{ 4 } \) , \(\frac { 1 }{ 2 } \) and - 6 \(\frac { 1 }{ 2 } \), respectively. Find the first term and the number of terms.

Find the number of terms common to the two AP's 3, 7, 11, .... 407 and 2, 9, 16, ....709.

Let S_n denote the sum of the first n terms of an A.P. If S_2n then prove that \(\frac { { S }_{ 6n } }{ { S }_{ 6n } } =\frac { 17 }{ 4 } \)

One side of an equilateral triangle is 18 em. The midpoints of its sides are joined to form another triangle whose midpoints, in term, are joined to form further another triangle and so on up to infinity. Fine the sum of the (i) Perimeters of all the triangle (ii) area of all the triangles.