Selina solutions for Concise Mathematics [English] Class 10 ICSE chapter 10 - Arithmetic Progression [Latest edition]

The first term and the common difference of an A.P. are 8 and –5 respectively. The A.P. is ______.

Is −8, −8, −8, −8, .......... an A.P.?

The 15th term of the A.P. 3, 0, –3, –6, ........ is ______.

The 24th term of an A.P. exceeds its 19th term by 10, its common difference is ______.

In the A.P. 8, 13, 18, ........., the nth term is 83, then n is equal to ______.

The n^th term of a sequence is (2n – 3), find its fifteenth term.

If the p^th term of an A.P. is (2p + 3); find the A.P.

Find the 24^th term of the sequence:

12, 10, 8, 6, .......

Find the 30^th term of the sequence:

`1/2, 1, 3/2,.........`

Find the \[100^{\text{th}}\] term of the sequence: \[\sqrt{5}, 2\sqrt{5}, 3\sqrt{5}, ..................\ .\] 

Find the 50^th term of the sequence:

`1/n, (n + 1)/n, (2n + 1)/n,..........`

Is 402 a term of the sequence:

8, 13, 18, 23, ................. ?  

Find the common difference and 99^th term of the arithmetic progression:

`7 3/4, 9 1/2, 11 1/4, ................` 

How many terms are there in the series 4, 7, 10, 13, ........,148?

How many terms are there in the series 0.5, 0.53, 0.56, ........, 1.1?

How many terms are there in the series `3/4, 1, 1 1/4, ........., 3`?

Which term of the A.P. 1, 4, 7, 10, ....... is 52?

If 5^th and 6^th terms of an A.P. are respectively 6 and 5, find the 11^th term of the A.P.

If t_n represents n^th term of an A.P., t₂ + t₅ – t₃ = 10 and t₂ + t₉ = 17, find its first term and its common difference.

Find the 10^th term from the end of the A.P. 4, 9, 14, .........., 254. 

Determine the arithmetic progression whose 3^rd term is 5 and 7^th term is 9.

Find the 31^st term of an A.P. whose 10^th term is 38 and the 16^th term is 74.

Which term of the series:

21, 18, 15 ....... is –81? 

Can any term of this series be zero? If yes, find the number of terms.

An A.P. consists of 60 terms, If the first and the last terms be 7 and 125 respectively, find the 31^st term.

The sum of the 4^th and the 8^th terms of an A.P. is 24 and the sum of the 6^th and the 10^th terms of the same A.P. is 34. Find the first three terms of the A.P.

If the third term of an A.P. is 5 and the seventh terms is 9, find the 17^th term.

Two A.P.s have the same common difference. If the difference between their 25th terms is 8, the difference between their 50th terms is ______.

Ten times the 10th term of an A.P. is equal to twenty times the 20th term of the same A.P. The 30th term of this A.P. is ______.

The nth term of an A.P. is 7n – 5. Its common difference is ______.

The 40th term of an A.P. exceeds its 16th term by 72. Then its common difference is ______.

The nth term of the A.P. 6, 11, 16, 21, ........ is 106, then the value of n – 4 is ______.

In an A.P., ten times of its tenth term is equal to thirty times of its 30^th term. Find its 40^th term.

How many two-digit numbers are divisible by 3?

Which term of the A.P. 5, 15, 25, ... will be 130 more than its 31st term?

Find the value of p, if x, 2x + p and 3x + 6 are in A.P.

If the 3^rd and the 9^th term of an A.P. be 4 and –8 respectively, find which term is zero?

How many three-digit numbers are divisible by 87?

For what value of n, the n^th terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... equal?

Determine the A.P. whose 3^rd term is 16 and the 7^th term exceeds the 5^th term by 12.

If numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n and its next two terms.

Determine the value of k for which k² + 4k + 8, 2k² + 3k + 6 and 3k² + 4k + 4 are in A.P.

State, true or false:

State, true or false:

If a, b and c are in A.P. then a + 4, b + 4 and c + 4 are in A.P.

An A.P. consists of 57 terms of which 7^th term is 13 and the last term is 108. Find the 45^th term of this A.P.

4^th term of an A.P. is equal to 3 times its first term and 7^th term exceeds twice the 3^rd term by 1. Find the first term and the common difference.

The sum of the 2nd term and the 7th term of an A.P. is 30. If its 15th term is 1 less than twice its 8th term, find the A.P.

In an A.P., if m^th term is n and n^th term is m, show that its r^th term is (m + n – r).

Which term of the A.P. 3, 10, 17, .......... will be 84 more than its 13^th term?

The sum of 41 terms of an A.P. with middle term 40 is ______.

The sum of all two digit numbers is ______.

The sum of A.P. 4, 7, 10, 13, ........ upto 20 terms is ______.

The sum of 40 terms of the A.P. 7 + 10 + 13 + 16 + .......... is ______.

The nth term of an A.P. is 6n + 4. The sum of its first 2 terms is ______.

How many terms of the A.P. : 24, 21, 18, ................ must be taken so that their sum is 78?

Find the sum of 28 terms of an A.P. whose n^th term is 8n – 5.

Find the sum of all odd natural numbers less than 50.

Find the sum of first 12 natural numbers each of which is a multiple of 7.

Find the sum of first 51 terms of an A.P. whose 2^nd and 3^rd terms are 14 and 18 respectively.

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

The first term of an A.P. is 5, the last term is 45 and the sum of its terms is 1000. Find the number of terms and the common difference of the A.P.

Find the sum of all natural numbers between 250 and 1000 which are divisible by 9.

The first and the last terms of an A.P. are 34 and 700 respectively. If the common difference is 18, how many terms are there and what is their sum?

In an A.P. the first term is 25, n^th term is –17 and the sum of n terms is 132. Find n and the common difference.

If 18, a, (b – 3) are in AP, then find the value of (2a – b).

Find an AP whose 4th term is 9 and the sum of its 6th and 13th terms is 40.

The sum of n natural numbers is 5n² + 4n. Find its 8^th term.

The fourth term of an A.P. is 11 and the eighth term exceeds twice the fourth term by 5. Find the A.P. and the sum of first 50 terms.

k + 2, 2k + 7 and 4k + 12 are the first three terms of an A.P. The first term of this A.P. is ______.

The sum of n terms of an A.P. is 3n². The second term of this A.P. is ______.

If 5, 7 and 9 are in A.P., then which of the following is in A.P.?

Find three numbers in A.P. whose sum is 24 and whose product is 440.

The angles of a quadrilateral are in A.P. with common difference 20°. Find its angles. 

Divide 96 into four parts which are in A.P. and the ratio between product of their means to product of their extremes is 15 : 7.

Find five numbers in A.P. whose sum is `12 1/2` and the ratio of the first to the last terms is 2 : 3.

Split 207 into three parts such that these are in A.P. and the product of the two smaller parts is 4623.

The sum of three numbers in A.P. is 15 and the sum of the squares of the extreme terms is 58. Find the numbers. 

Find four numbers in A.P. whose sum is 20 and the sum of whose squares is 120.

Insert one arithmetic mean between 3 and 13.

The angles of a polygon are in A.P. with common difference 5°. If the smallest angle is 120°, find the number of sides of the polygon.

If the pth term of an AP is q and its qth term is p then show that its (p + q)th term is zero.

If \(a, b\) and \(c\) are \(p^{\text {th }}, q^{\text {th }}\) and \(r^{\text {th }}\) terms of an A.P., prove that:

\[ a(q-r)+b(r-p)+c(p-q)=0 \]

Show that a², b², c² are in A.P., if \[\frac{1}{b + c}, \frac{1}{c + a}, \frac{1}{a + b}\] are in A.P.

A man saved ₹ 7,65,000 in 10 years. In each year, after the first, he saved ₹ 6,000 more than he did in the preceding year. How much did he save in the first seven years.

18^th term of an A.P. is equal to 4 times its 4^th term and the 6^th term exceeds twice the 2^nd term by 4. Find the sum of the first 9 terms of this A.P.

If A, B and C are three arithmetic progressions (APs) as given below:

A = 2, 4, 6, 8, ....... upto n terms 

B = 3, 6, 9, 12, ....... upto n terms 

C = 0, 4, 8, 12, ....... upto n terms, then out of A + B, A − C, C − B and B – А which is/are A.P.?

In an A.P., a = −36, d = 18 and l = 36, then n is ______.

Do the numbers \(1^{2}, 5^{2}, 7^{2}, 73 \ldots\) form an A.P.? If yes, its next term will be:

The sum of first 10 even natural numbers is ______.

For the given numbers \(\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \cdots\)

Assertion (A): To find whether these terms form an A.P. or not. Express each term as the product of a natural number and \(\sqrt{3}\)
i.e. \(\sqrt{3}=1 \times \sqrt{3}, \sqrt{12}=2 \sqrt{3}\),
\(\sqrt{27}=3 \sqrt{3}, \sqrt{48}=4 \sqrt{3}\), etc.

Reason (R): Since, for the given numbers difference between the consecutive terms is same. It is an A.P.

\[5, 8, 11, 14, .....\] are in A.P.

Assertion (A): \[\frac{5}{2}, 4, \frac{11}{2}, 7, .....\] are also in A.P.

Reason (R): If each term of a given A.P. is divided by the same non-zero number, the resulting sequence is an A.P.

An A.P. with 3^rd term = −8 and 9^th term = 4. 

Assertion (A): Common difference, d = −2. 

Reason (R): If first term of the A.P. is a, then (a + 8d) − (a + 2d) = 4 + 8.

The n^th term of a sequence = `5 n^2 - 3`.

Statement (1): The sequence is an A.P.

Statement (2): If the n^th term of a sequence is not linear; the sequence does not form an A.P.

The sum of first ten terms of an A.P. = 3 and sum of its first fifteen terms = 16.

Statement (1): The sum of first 5 terms of the given A.P. = 16 − 3 = 13.

Statement (2): The sum of last 5 terms of the given A.P. = Sum of first 15 terms minus sum of first 10 terms.

The 6^th term of an A.P. is 16 and the 14^th term is 32. Determine the 36^th term.

If the 3^rd and the 9^th term of an A.P. be 4 and –8 respectively, find which term is zero?

An A.P. consists of 50 terms of which 3^rd term is 12 and the last term is 106. Find the 29^th term of the A.P.

Find the sum of first 20 terms of an A.P. whose first term is 3 and the last term is 57.

How many terms of the series 18 + 15 + 12 + ........ when added together will give 45?

Find the general term (n^th term) and 23^rd term of the sequence 3, 1, –1, –3, ........... . 

Is –150 a term of 11, 8, 5, 2, .......?

How many multiples of 4 lie between 10 and 250?

The 25^th term of an A.P. exceeds its 9^th term by 16. Find its common difference.

If the n^th term of the A.P. 58, 60, 62, .... is equal to the n^th term of the A.P. –2, 5, 12, …., find the value of n.

Which term of the A.P. 105, 101, 97, ........  is the first negative term?

Divide 216 into three parts which are in A.P. and the product of two smaller parts is 5040.

Can 2n² – 7 be the n^th term of an A.P.? Explain.

The first term of an A.P. is 20 and the sum of its first seven terms is 2100; find the 31^st term of this A.P.

Find the sum of last 8 terms of the A.P. –12, –10, –8, ……, 58.

An A.P. consists of 57 terms of which 7^th term is 13 and the last term is 108. Find the 45^th term of this A.P.

Ten times the tenth term of an A.P. is equal to fifteen times, its fifteenth term. Find the twenty-fifth term of this A.P.

The sum of the 2nd term and the 7th term of an A.P. is 30. If its 15th term is 1 less than twice its 8th term, find the A.P.

Refer the given sequence \[23, 21\frac{1}{2}, 20, ...\]

1. Find the general term of the given sequence.
2. Which term is the last positive term in the sequence.

Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of the cable car ride from the base point to the top-most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which the car moves.

The distance of the first pole from the base point is 200 m and subsequent poles are installed at an equal interval of 150 m. Further, the distance of the last pole from the top is 300 m.

Based on the above information, answer the following questions using arithmetic progression:

1. Find the distance of the 10^th pole from the base. 
2. Find the distance between the 15^th pole and the 25^th pole. 
3. Find the time taken by the cable car to reach the 15^th pole from the top if it is moving at the speed of 5 m/sec and coming from the top. 

A school auditorium is to be constructed to accommodate at least 1500 people. The chairs are to be placed in a concentric circular arrangement in such a way that each succeeding circular row has 10 seats more than the previous one.

(i) If the first circular row has 30 seats, how many seats will the \(10^{\text {th }}\) row have?

(ii) For 1,500 seats in the auditorium, how many circular rows need to be there?

(iii) If there were 17 rows in the auditorium, how many seats will be there in the middle row?

The figure shows a big triangle in which multiple other triangles can be seen. Observe the pattern of dark shaded and light unshaded triangles starting with one triangle in row 1, three triangles in row 2, five triangles in row 3 and so on.

Based on the above information, answer the following questions:

1. How many triangles will be there in the 15^th row?
2. In which row will the number of triangles be 47?
3. The number of dark shaded triangles in each row are in A.P. Find the total number of dark shaded triangles in the first 15 rows.