If the first term of an AP. is 5 and common difference is - 3 then sum of its 60 terms is equal to ______.

The three geometric means between the numbers 1 and 81 are ______.

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 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 the Sum of first 20 terms of an AP, n whichg 3rd terms is 7 and 7th term is two more than thrice of its 3rd term.

Find the 20th term of the series;
2 x 4 + 4 x 6 + 6 x 8 + ... + n terms

Find the nth term of the series 1 + 2 + 4 + 7 +......

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.

Find the sum to infinity of the G.P  6,1.2,.24 

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

Find the indicated terms in each of the sequence, where nth terms are given.
(i) b_n = \({ (-1) }^{ n }({ n }^{ 2 }-1)\), b_7, b₁₃ 
(ii) \(a_{n}=\frac{n^{2}}{2^{n}}, a_{7}\)

If the mth term  of an AP be \({ 1 }/{ n }\) and its nth term be \({ 1 }/{ m }\) , then show that its mnth term is 1.

At the end of each yearthe value of a certain machine has depreciated by 20% of its value at the begining of that year. If its initial value was Rs.1250, then find the value at the end of 5 yrs.

If a is the A.M of band c and the two G.M's C₁ and C₂ then prove that C ₁³ + C₂ ³ = 2abc

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

Find the 15th term from the end of the AP 3, 5, 7, 9, ......, 201.

If the first and the n^th term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that p² = (ab)ⁿ.

If the sum of a certain number of terms of the A.P. 25, 22, 19, .... is 116, find the last term.

If a, b, c are A .P and A₁ is the AM of a and band A₂ is the AM. of band c then prove that the AM of A₁ and A₂ is b

A manufacturer reckons that the value of a machine, which costs him Rs. 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.

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.