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Find the sum of the following series to n terms:
13 + 33 + 53 + 73 + ……..
23 + 43 + 63 + 83 + ………
1.2.5 + 2.3.6 + 3.4.7 + ……..
1.2.4 + 2.3.7 + 3.4.10 + …………
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………
1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ………….
3 × 12 + 5 × 22 + 7 × 32 + …………..
Find the sum of the series whose nth term is :
2n3 + 3n2 – 1
n3 – 3n
n(n + 1) (n + 4)
(2n – 1)2
Find the 20th term and the sum of 20 terms of the series :
2 × 4 + 4 × 6 + 6 × 8 + ……………..
If the line segment joining the points P(x1, y1) and Q(x2, y2) subtends an angle α at the origin O, prove that : OP. OQ cos α = x1 x2 + y1 y2.
The vertices of a triangle ABC are A(0, 0), B (2, -1) and C (9, 0). Find cos B.
Four points A (6, 3), B(-3, 5), C(4, -2) and D(x, 3x) are given in such a way that, find x.
The points A (2, 0), B(9, 1), C (11, 6) and D (4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.
Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are (-36, 7), (20, 7) and (0, -8).
The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find the vertices of the triangle.
Find the distance between P(x1, y1) and Q(x2, y2) when (i) PQ is parallel to the y-axis (ii) PQ is parallel to the x-axis.
Find a point on the x-axis, which is equidistant from the point (7, 6) and (3, 4).
Sum the following series to n terms :
3 + 5 + 9 + 15 + 23 + ………….
2 + 5 + 10 + 17 + 26 + ………..
1 + 3 + 7 + 13 + 21 + …………….
3 + 7 + 14 + 24 + 37 + ……………..
1 + 3 + 6 + 10 + 15 + …………….
1 + 4 + 13 + 40 + 121 + ……….
4 + 6 + 9 + 13 + 18 + ………..
2 + 4 + 7 + 11 + 16 + ………….
Find the locus of a point equidistant from the point (2, 4) and the y-axis.
Find the equation of the locus of a point which moves such that the ratio of its distance from (2, 0) and (1, 3) is 5 : 4.
A point moves as so that the difference of its distances from (ae, 0) and (-ae, 0) is 2a, prove that the equation to its locus is , where b2 = a2(e2 – 1).
Find the locus of a point such that the sum of its distances from (0, 2) and (0, -2) is 6.
Find the locus of a point which is equidistant from (1, 3) and x-axis.
Find the locus of a point which moves such that its distance from the origin is three times is the distance from the x-axis.
A(5, 3), B(3, -2) are two fixed points, find the equation to the locus of a point P which moves so that the area of the triangle PAB is 9 units.
Find the locus of a point such that the line segments having end points (2, 0) and (-2, 0) subtend a right angle at that point.
If A (-1, 1) and B (2, 3) are two fixed points, find the locus of a point P so that the area d ΔPAB = 8 sq. units.
A rod of length l slides between the two perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1 : 2.
Find the locus of the mid-point of the portion of the x cos α + y sin α = p which is intercepted between the axes.
If O is the origin and Q is a variable point on y2 = x, Find the locus of the mid-point of OQ.
Write the sum of the series : 2 + 4 + 6 + 8 + ….. + 2n
Write the sum of the series : 12 – 22 + 32 – 42 + 52 – 62 + ……. + (2n – 1)2 – (2n)2
Write the sum to n terms of a series whose rth term is: r + 2r
If , find .
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then write the value of k.
Write the sum of 20 terms of the series :
Write the 50th term of the series 2 + 3 + 6 + 11 + 18 + ………
Let Sn denote the sum of the cubes of first n natural numbers, and sn denote the sum of first n natural numbers. Then write the value of
The sum to n terms of the series
The sum of the series :
The value of is equal to
If , then
If , then Sn is equal to
If to n terms is S. Then, S is equal to
Sum of n terms of the series is
The sum of 10 terms of the series is
The sum of the series 12 + 32 + 52 + …… to n terms is
The sum of the series to n terms is
