Arithmetic Progression (hots) question class -Xth

 Maths Hots Question

Arithmetic Progression

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1. If the pth term of an A.P is q and qth term is p ,prove that nth term is (p+q-n).

2. If m times the mth term of an A.P is equal to n times it's nth term,show that the(m+n)the term of an AP is zero.

3. If pth,qth and rth terms of an A.P are a,b,c respectively ,then show that

i.a(q-r)+b(r-p)+c(p-q)=0

ii.(a-b)r +(b-c)p +(c-a)q=0

4. How many numbers of two digits are divisible by 7?

5. Find the number of integers between 50 and 500 which are divisible by 7.

6 .Which terms of the A.P- 3,15,27,39.....will be 132 more than it's 54th term?

7. Two  A.Ps have the same common difference .The first term of one of these is 3, and that of the other is 8. What is the difference between their 

i.2nd terms ?

ii.4th terms?

iii.10th term?

iv.30th term?

8. The sum of three numbers in A.P is -3 ,and their product is 8. Find the numbers.

9. Divide 32 into four parts which are in A.P.

such that the product of extremes is to the product of means is 7:15.

10. If the .numbers a,b,c,d,e form an AP .,then find the value of a -4b + 6c - 4d + e.

11. If a^n+1 +b^n+1/a^n + b^n is the A.M between and b ,Then find the value of n.

12. The sum of first six terms of an arithmetic progression is 42.The ratio of it's 10th term to it's 30th term is 1:3 . Calculate the first and the thirteenth term of A.P.

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13. Find the sum of all three digit natural numbers ,which are divisible by 7?

14.Find the sum of all natural numbers between 250 and 1000 which are exactly divisible by 3.

15. Find the number of terms in the series 20 + 19  1/3  + 18  2/3 +........ of which the sum is 300,explain the double answer .

16. In an AP .,the sum of first n terms is 3n^2/2 +5n/2 .Find it's 25th term.

17.The sum of the third and seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the A.P.

18.Raghav buys a shop for Rs 1,20,000. He pays half of the amount in cash and agrees to pay the balance in 12 annual instalments of Rs 5000 each .If the rate of interest is 12% and he pays with the instalments the interest due on the unpaid amounts,find the total cost of the shop.

19.The digits of a a positive integer, having three digits 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.

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

21. Along a road lie an odd number of stones placed at intervals of 10metres .These stones have to be assembled around middle stone. A person can carry only one stone at a time .A man carried the job with one of the end stones by carrying them in succession .In carrying all the stones he covered a distance of 3 km .Find the number of stones.

22. A ladder has rungs 25cm apart .The rungs decrease uniformly in length from 45 cm at the bottom to 25cm at the top.If the top and bottom rungs are 2.5,metre apart ,what is the length of the wood required for the rungs.

23.A small terrace at a football ground comprises of  15 steps each of which is 50m long and built of solid concrete .Each steps has a rise of 1/4 m and a tread of 1/2 m.Calculate the total volume of concrete required to build the terrace.

24. If the 8th term of an A.p is 31 and the 15th term is 16 more than the 11th term,find the A.P.

25. Which term of arithmetic progression 5,15,25......... will be 130 more than it's 31st term?

26.If the 10th term of an AP is 52 and 17th term is 20 more than the 13th term, find the AP.

27.The 10th term of an AP is 52 and 16th term is 82. Find the 32nd term and the general term.

28.The sum of 5th and 9th term of an AP is 72 and the sum of  7th and 12th term is 97. 

Find the AP.

29.The sum of the third and seventh term of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the A.P.

30.If the mth term of an AP is 1/n and the nth term is 1/m,  show that the sum of mn term is 1/2 (mn+1).

31. The sum of n ,2n and 3n of an AP are S1 ,S2 and S3 respectively. Then prove that---

S3 =3(S2-S1).

32.The sum of n terms of three arithmetical progressions are S1, S2 and S3. The first term of each is unity and the common difference are 1 ,2, and 3 respectively .Prove that

S1 + S2 = 2S2.

33.If in an AP the sum of m terms is equal to n and the sum of n terms is equal to m ,then prove that the sum of (m+n) terms is -(m+n).

34.The ratio of the sum of n terms of two A.Ps is (7n+1) : (4n +27).Find the ratio of their mth terms .

35.The ratio of the sums of m and n terms of an AP is m^2:n^2. Show that the ratio of the mth and nth term is (2m-1):(2n-1).

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By Anuranjan Gadekar

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