Write first 4 terms in each of the sequences:

(i) a_{n =} (5n + 2)

Find the first five terms of the sequence, defined by

a₁ = 1, a_n = a_n–1 + 3 for n ≥ 2.

Find the first 5 terms of the sequence, defined by

a₁ = –1, a_n = for n ≥ 2.

Find the 23^rd term of the AP 7, 3, 1, –1, –3, …

Find the 20^th term of the AP , 3, 5, 7 , ….

Find the n^th term of the AP 8, 3, –2, –7, –12, ….

Find the n^th term of the AP 1, , ….

Which term of the AP 9, 14, 19,24, 29, …. is 379?

Which term of the AP 64, 60, 56, 52, 48, …. is 0?

How many terms are there in the AP 11, 18, 25, 32, 39, …. 207?

How many terms are there in the AP 1, 1 , …., –16 ?

Is - 47 a term of the AP 5, 2, –1, –4, –7, ….?

The 5^th and 13^th terms of an AP are 5 and –3 respectively. Find the AP and its 30^th term.

The 2^nd, 31^st and the last terms of an AP are 7 and –6 respectively. Find the first term and the number of terms.

If the 9^th term of an AP is 0, prove that its 29^th term is double the 19^th term.

The 4^th term of an AP is three times the first and the 7^th term exceeds twice the third term by 1. Find the first term and the common difference.

If 7 times the 7^th term of an AP is equal to 11 times its 11^th term, show that the 18^th term of the AP is zero.

Find the 28^th term from the end of the AP 6, 9, 12, 15, 18, …., 102.

Find the 16^th term from the end of the AP 7, 2, –3, –8, –13, …., –113

How many 3 - digit numbers are divisible by 7?

How many 2 - digit numbers are divisible by 3?

If θ₁, θ₂, θ₃, …., θ_n are in AP whose common difference is d, show that

sec θ₁sec θ₂ + sec θ₂sec θ₃ + …. + sec θ_n–1sec θn = .

In an AP, it is being given that . Find .

Three numbers are in AP. If their sum is 27 and their product is 648, find the numbers.

The sum of three consecutive terms of an AP is 21, and the sum of the squares of these terms is 165. Find these terms

The angles of a quadrilateral are in AP whose common difference is 10°. Find the angles.

The digits of a 3 - digit number are in AP, and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.

Find the number of terms common to the two arithmetic progressions 5, 9, 13, 17, …., 217 and 3, 9, 15, 21, …., 321.

We know that the sum of the interior angles of a triangle is 180°. Show that the sum 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 24 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. Find the perimeter of the sixth inscribed equilateral triangle.

A man starts repaying a loan as the first instalment of 10000. If he increases the instalment by 500 every month, what amount will he pay in 30^th instalment?

Find the sum of 23 terms of the AP 17, 12, 7, 2, –3, ….

Find the sum of 16 terms of the AP 6, 5 , 4 , 4, ….

Find the sum of 25 terms of the AP , 2, 3, 4, ….

Find the sum of 100 term of the AP 0.6, 0.61, 0.62, 0.63, ….

Find the sum of 20 terms of the AP (x + y), (x – y), (x – 3y), ….

Find the sum of n term of the AP , ….

Find the sum of the series 2 + 5 + 8 + 11 + …. + 191.

Find the sum of the series 101 + 99 + 97 + 95 + …. + 43.

Find the sum of the series 1 + 4 + 7 + 10 + …. + x = 715.

Find the value of x such that 25 + 22 + 19 + 16 + …. + x = 112.

Find the r^th term of the AP, the sum of whose first n terms is (3n² + 2n).

Find the sum of n term of an AP whose r^th term is (5r + 1).

If the sum of a certain number of terms of the AP 27, 24, 21, 18, …. is –30, find the last term.

How many terms of the AP 26, 21 16, 11, …. are needed to give the sum 11?

How many terms of the AP 18 16, 14, 12, …. are needed to give the sum 78? Explain the double answer.

How many terms of the AP 20, must be taken to make the sum 300? Explain the double answer.

Thesumsof an terms of two arithmetic progressions are in the ratio (7n – 5) : (5n + 17). Show that their 6^th terms are equal.

If the ratio between the sums of n terms of two arithmetic progressions is (7n + 1) : (4n + 27), find the ratio of their 11^th terms.

Find the sum of all odd integers from 1 to 201.

Find the sum of all even integers between 101 and 199.

Find the sum of all integers between 101 and 500, which are divisible by 9.

Find the sum of all integers between 100 and 600, each of which when divided by 5 leaves 2 as remainder.

The sum of first 7 terms of an AP is 10 and that of next 7 terms is 17. Find the AP.

If the sum of n terms of an AP is (3n² + 5n) and its m^th term is 164, find the value of m.

Find the sum of all natural numbers from 1 and 100 which are divisible by 4 or 5.

If the sum of n terms of an AP is , where P and Q are constants then find the common difference.

If S_m = m²p and S_n = n²p, where m ≠ n in an AP then prove that S_p = p³.

A carpenter was hired to build 192 window frames. The first day he made 5 frames and each day, thereafter he made 2 more frames than he made the day before. How many days did he take to finish the job?

The interior angles of a polygon are in AP. The smallest angle is 52^0, and the common difference is 8⁰. Find the number of sides of the polygon.

A circle is completely divided into n sectors in such a way that the angles of the sectors are in AP. If the smallest of these angles is 8⁰ and the largest is 72⁰, calculate n and the angle in the fifth sector.

There are 30 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A Gardner waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the Gardner will cover in order to water all the trees.

Two cars start together from the same place in the same direction. The first go with a uniform speed of 60 km/hr. The second goes at a speed of 48 km/hr in the first hour and increases the speed by 1 km each succeeding hour. After how many hours will the second car overtake the first car if both cars go non - stop?

Arun buys a scooter for ₹44000. He pays ₹8000 in cash and agrees to pay the balance in annual instalments of ₹4000 each plus 10% interest on the unpaid amount. How much did he pay for it?

A man accepts a position with an initial salary of ₹26000 per month. It is understood that he will receive an automatic increase of ₹250 in the very next month and each month thereafter.

Find this (i) salary for the 10^th month, (ii) total earnings during the first year.

A man saved ₹660000 in 20 years. In each succeeding year after the first year, he saved ₹2000 more than what he saved in the previous year. How much did he save in the first year?

150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day, and so on. It takes 8 more days to finish work now. Find the number of days in which the work was completed.

A man saves ₹4000 during the first year, ₹5000 during the second year and in this way he increases his savings by ₹1000 every year. Find in what time his savings will be ₹85000.

A man arranges to pay off a debt of ₹36000 by 40 annual instalments which form an AP. When 30 of the instalments are paid, he dies, leaving one - third of the debt unpaid. Find the value of the first instalment.

A manufacturer of TV sets produced 6000 units in the third year and 7000 units in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the production

(i) in the first year,

(ii) in the 10^th year,

(iii) in 7 years.

A farmer buys a used for ₹180000. He pays ₹90000 in cash and agrees to pay the balance in annual instalments of ₹9000 plus 12% interest on the unpaid amount. How much did the tractor cost him?

Find the arithmetic mean between:

(i) 9 and 19

(ii) 15 and -7

(iii) -16 and -8

Insert four arithmetic means between 4 and 29.

Insert three arithmetic means between 23 and 7.

Insert six arithmetic means between 11 and -10.

There is n arithmetic means between 9 and 27. If the ratio of the last mean to the first mean is 2 : 1, find the value of n.

Insert arithmetic means between 16 and 65 such that the 5^th AM is 51. Find the number of arithmetic means.

Insert five numbers between 11 and 29 such that the resulting sequence is an AP.

Prove that the ratio of sum of m arithmetic means between the two numbers to the sum of n arithmetic means between them is m:n.

If a, b, c are in AP, prove that

(i) (a – c)² = 4(a – b)(b – c)

(ii) a² + c² + 4ac = 2(ab + bc + ca)

(iii) a³ + c³ + 6abc = 8b³

If a, b, c are in AP, show that

(a + 2b – c)(2b + c – a)(c + a – b) = 4abc.

If a, b, c are in AP, show that

(i) (b + c – a), (c + a – b), (a + b – c) are in AP.

(ii) (bc – a²), (ca – b²), (ab – c²) are in AP.

If are in AP, prove that

(i) are in AP.

(ii) are in AP.

If are in AP, prove that a²(b + c), b²(c + a), c²(a + b) are in AP.

If a, b, c are in AP, show that are also in AP.

If the sum of n terms of an AP is given by S_n = (2n² + 3n), then find its common difference.

If 9 times the 9^th term of an AP is equal to 13 times the 13^th term, show that its 22^nd term is 0.

In an AP it is given that S_n = qn² and S_m = qm². Prove that S_q = q³.

Find three arithmetic means between 6 and - 6.

The 9^th term of an AP is 0. Prove that its 29^th term is double the 19^th term.

How many terms are there in the AP 13, 16, 19, …., 43?

Find the 8^th term from the end of the AP 7, 9, 11, …., 201.

How many 2 - digit numbers are divisible by 7?

If 7^th and 13^th terms of an AP be 34 and 64 respectively then find its 18^th term.

What is the 10^th common term between the APs 3, 7, 11, 15, 19, … and 1, 6, 11, 16, …?

The first and last terms of an AP are 1 and 11 respectively. If the sum of its terms is 36, find the number of terms.

In an AP, the p^th term is q and (p + q)^th term is 0. Show that its q^th term is p.

Write the sum of first n even natural numbers.

Write the sum of first n odd natural numbers.

The sum of n terms of an AP is. Find the common difference.

If the sums of n terms of two APs are in ratio (2n + 3) : (3n + 2), find the ratio of their 10^th terms.