## Arithmetic Progressions

**Multiple Choice Questions**

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

(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**

**Ans:(B)**

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

**Ans:(B)**

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

**Ans:(C)**

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

**★★★★★★**

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.

**ANSWER:**

**Work in progress.**

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