NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 9 - Sequences and Series [Latest edition]

The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.

The p^th term of an A.P. is a and q^th term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.

If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n

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

Find the sum of first 24 terms of the A.P. a₁, a₂, a₃, ... if it is known that a₁ + a₅ + a₁₀ + a₁₅ + a₂₀ + a₂₄ = 225.

The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers

Show that (x² + xy + y²), (z² + xz + x²) and (y² + yz + z²) are consecutive terms of an A.P., if x, y and z are in A.P.

If a, b, c, d are in G.P., prove that a² – b², b² – c², c² – d² are also in G.P.

If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`

If a₁, a₂, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a₂ + cosec a₂ cosec a₃ + ...+ cosec a_n–1 cosec a_n) is equal to cot a₁ – cot a_n 

If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad

If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c

If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. Then prove that x^{b – c}. y^{c – a} . z^{a – b} = 1

Find the natural number a for which ` sum_(k = 1)^n f(a + k)` = 16(2ⁿ – 1), where the function f satisfies f(x + y) = f(x) . f(y) for all natural numbers x, y and further f(1) = 2.

A sequence may be defined as a ______.

If x, y, z are positive integers then value of expression (x + y)(y + z)(z + x) is ______.

In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.

In an A.P. the p^th term is q and the (p + q)^th term is 0. Then the q^th term is ______.

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P² R³ : S³ is equal to ______.

The 10th common term between the series 3 + 7 + 11 + ... and 1 + 6 + 11 + ... is ______.

In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.

The minimum value of the expression 3^x + 3^1–x, x ∈ R, is ______.

The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`

A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?

A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month

A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. What is his total earnings during the first year?

If the p^th and q^th terms of a G.P. are q and p respectively, show that its (p + q)^th term is `(q^p/p^q)^(1/(p - q))`

A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?

We know the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.

A side of an equilateral triangle is 20 cm long. A second equilateral triangle is inscribed in it by joining the midpoints of the sides of the first triangle. The process is continued as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.

In a potato race 20 potatoes are placed in a line at intervals of 4 metres with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?

In a cricket tournament 16 school teams participated. A sum of Rs 8000 is to be awarded among themselves as prize money. If the last-placed team is awarded Rs 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first-place team receive?

If a₁, a₂, a₃, ..., an are in A.P., where ai > 0 for all i, show that `1/(sqrt(a_1) + sqrt(a_2)) + 1/(sqrt(a_2) + sqrt(a_3)) + ... + 1/(sqrt(a_(n - 1)) + sqrt(a_n)) = (n - 1)/(sqrt(a_1) + sqrt(a_n))`

Find the sum of the series (3³ – 2³) + (5³ – 4³) + (7³ – 6³) + … to n terms

Find the sum of the series (3³ – 2³) + (5³ – 4³) + (7³ – 6³) + ... to 10 terms

Find the r^th term of an A.P. sum of whose first n terms is 2n + 3n² 

If A is the arithmetic mean and G₁, G₂ be two geometric means between any two numbers, then prove that 2A = `(G_1^2)/(G_2) + (G_2^2)/(G_1)`

If θ₁, θ₂, θ₃, ..., θ_n are in A.P., whose common difference is d, show that secθ₁ secθ₂ + secθ₂ secθ₃ + ... + secθ_n–1 . secθ_n = `(tan theta_n - tan theta_1)/sin d`

If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).

If p^th, q^th, and r^th terms of an A.P. and G.P. are both a, b and c respectively, show that a^b–c . b^{c – a} . c^{a – b} = 1

If the sum of n terms of an A.P. is given by S_n = 3n + 2n², then the common difference of the A.P. is ______.

The third term of G.P. is 4. The product of its first 5 terms is ______.

If 9 times the 9^th term of an A.P. is equal to 13 times the 13^th term, then the 22^nd term of the A.P. is ______.

If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.

If in an A.P., S_n = qn² and S_m = qm², where S_r denotes the sum of r terms of the A.P., then Sq equals ______.

Let S_n denote the sum of the first n terms of an A.P. If S_2n = 3S_n then S_3n: S_n is equal to ______.

The minimum value of 4^x + 4^1–x, x ∈ R, is ______.

Let S_n denote the sum of the cubes of the first n natural numbers and s_n denote the sum of the first n natural numbers. Then `sum_(r = 1)^n S_r/s_r` equals ______.

If t_n denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + ... then t₅₀ is ______.

The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm³ and the total surface area is 252cm². The length of the longest edge is ______.

For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.

The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.

The third term of a G.P. is 4, the product of the first five terms is ______.

Two sequences cannot be in both A.P. and G.P. together.

Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.

Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.

The sum or difference of two G.P.s, is again a G.P.

If the sum of n terms of a sequence is quadratic expression then it always represents an A.P