Arithmetic Progression Question Bank

Arithmetic Progressions 

Multiple Choice Questions

1. In an AP, if  `d = -4, n = 7, a_n = 4` then a is 

(A) 6 
(B) 7 
(C) 20
(D) 28

Ans: (D)

2. In an AP, if `a = 3.5, d = 0, n = 101`, then `a_n` will be

(A) 0 

(B) 3.5

(C) 103.5 

(D) 104.5

Ans:(B)

3. The list of numbers – 10, – 6, – 2, 2,... is

(A) an AP with d = – 16

(B) an AP with d = 4

(C) an AP with d = – 4

(D) not an AP

Ans:(B)

4. The first four terms of an AP, whose first term is –2 and the common difference is –2, are

(A) – 2, 0, 2, 4

(B) – 2, 4, – 8, 16

(C) – 2, – 4, – 6, – 8

(D) – 2, – 4, – 8, –16

Ans:(C)

5. The 11th term of the AP: `–5, –5/2, 0, 5/2`, ...is

(A) –20

(B) 20 

(C) –30

(D) 30

Ans: 

★★★★★★

1. Find a and b such that the numbers a, 9, b, 25 forms an AP.

2. If the seventh term of an AP is 1/9 and its ninth term is 1/7, find It's 63rd term.

3. The sum of the 4th and 8th terms of an AP is 24 and the sum of its 6th and 10th term is 44. Find the first three terms of the AP.

4. Which term of the AP 3, 15, 27, 39..... will be 120 more than its 21st term?

5. Is 51 a term of the AP 5, 8, 11, 14....?

6. How many terms are there in the AP 7, 11, 15, ......139?

7. Find the middle term of AP 213, 205, 197, ........,37.

8. Which term of the AP 24, 21, 18, 15, ......is the first negative term?

9. For what value of n are the nth terms of the following two APs the same 13, 19, 25, ....and 69, 68, 67, ....? Also, find these terms.

10. If seven times the 7th term of an AP is equal to eleven times the 11th term then what will be its 18th term?

11. If the nth term of a progression be a linear expression in n then prove that this progression is an AP.

12. In a given AP if pth term is q and the qth term is p then show that the nth term is (p + q - n).

13. If m times the mth term of an AP is equal to n times the nth term and m≠n, show that its (m+n)th term is zero.

14. Show that the sum of an AP whose first term is `a`, the second term `b` and the last term `c`, is equal to

`frac{(a + c)(b+ c - 2a)}{2(b - a)}`

15. Solve the equation `– 4 + (–1) + 2 +...+ x = 437`

16. The ratio of the `11^{th}` term to the `18^{th}` term of an AP is `2 : 3`. Find the ratio of the `5^{th}` term to the `21^{st}` term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

17. If `S_n`  denotes the sum of first `n` terms of an AP, prove that `S_{12} = 3(S_8 –S_4 )`

Case Studies 

1. Push-ups are a fast and effective exercise for building strength. These are helpful in almost all sports including athletics. While the push-up primarily targets the muscles of the chest, arms, and shoulders, support required from other muscles helps in toning up the whole body.

Nitesh wants to participate in the push-up challenge. He can currently make 3000 push-ups in one hour. But he wants to achieve a target of 3900 push-ups in 1 hour for which he practices regularly. With each day of practice, he is able to make 5 more push-ups in one hour as compared to the previous day. If on the first day of practice he makes 3000 push-ups and continues to practice regularly till his target is achieved. Keeping the above situation in mind answer the following questions:

i) Form an A.P representing the number of push-ups per day and hence find the minimum number of days he needs to practice before the day his goal is accomplished?

ii) Find the total number of push-ups performed by Nitesh up to the day his goal is achieved.

2. India is a competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in `6^{th}` year and 22600 in `9^{th}` year.

1. Find the production during the first year.

2. Find the production during the `8^{th}` year.

3. Find the production during the first 3 years.

4. In which year, the production is  ₹29,200.

5. Find the difference in the production during `7^{th}` year and the `4^{th}` year

ANSWER:

1.  ₹5000

2. Production during `8^{th}` year is `(a+7d)= 5000 + 2(2200) =20400` 

3. Production during first 3 year `= 5000 + 7200 + 9400=21600`

4. `N=12`

5. Difference `=  18200-11600=6600`

3. Your friend Veer wants to participate in a 200m race. He can currently run that distance in 51 seconds and with each day of practice, it takes him 2 seconds less. He wants to do in 31 seconds.

1. What is the minimum number of days he needs to practice till his goal is achieved.

2. If `n^{th}` term of an AP is given by `a_n = 2n + 3` then the common difference of an AP is...

3. The value of `x`, for which `2x, x+ 10, 3x + 2` are three consecutive terms of an AP

4. Your elder brother wants to buy a car and plans to take loan from a bank for his car. He repays his total loan of ₹1,18,000 by paying every month starting with the first instalment of  ₹1000. If he increases the instalment by ₹100 every month , answer the following:

1. The amount paid by him in 30th instalment is........

2. The amount paid by him in the 30 installments is...

3. What amount does he still have to pay offer 30th installment?

4. If total installments are 40 then amount paid in the last installment?

5. The ratio of the 1st installment to the last installment is...

 Work in progress.