Class X Arithmetic Progression

Based on Last 10 year CBSE Question Paper 

1. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint : Sx-1 = S49 – Sx ]

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

3. A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.

4. If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is S1)? What is the sum of the first two terms? What is the second term? Similarly find the 3rd, the 10th and the nth terms.

5.  Find the sum of the first 22 terms of an AP in which d = 7 and 22nd term is 149.

6. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

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

8. Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her weekly savings become Rs 20.75, find n.

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

10. How many multiples of 4 lie between 10 and 250?

11. Which term of the AP 3, 15, 27, 39, … will be 132 more than its 54th term?

12. If the 3rd and the 9th terms of an AP are 4 and  -8, respectively, then which term of this AP is zero.

13. Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0?

14. Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . .

15. The angles of a triangle are in A.P., the least being half the greatest. Find the angles. 

16. Which term of the progression 20, 192, 183, 17 … is the first negative term?

17. The 4th term of an A.P. is zero. Prove that the 25th term of the A.P. is three times its 11thterm

18. The 7th term of an A.P. is 20 and its 13th term is 32. Find the A.P.

19. Find how many two-digit numbers are divisible by 6?

20. How many natural numbers are there between 200 and 500, which are divisible by 7?

21. Find the number of all three-digit natural numbers which are divisible by 9.

22. Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

23. Find the middle term of the A.P. 6, 13, 20, …, 216.

24. How many terms of the A.P. 65, 60, 55, … be taken so that their sum is zero?

25. Find the sum of the first 25 terms of an A.P. whose nth term is given by tn = 2 – 3n

26. The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference and number of terms.

27. In an AP, if S5 + S7 = 167 and S10 = 235, then find the AP, where s, denotes the sum of its first n terms.

28. Find the sum of all three digit natural numbers, which are multiples of 11.

29. Which term of the A.P. 3, 14, 25, 36, … will be 99 more than its 25th term?

30. The 19th term of an AP is equal to three times its 6th term. If its 9th term is 19, find the A.P. 

31. The 14th term of an AP is twice its 9th term. If its 6th term is -8, then find the sum of its first 20 terms.

32. The sum of first n terms of an AP is 3n2 + 4n. Find the 25th term of this AP.

33.  The sum of the first seven terms of an AP is 182. If its 4th and the 17th terms are in the ratio 1 : 5, find the AP.

34. If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3[S20 – S10]

35. The digits of a positive number of three digits are in A.P. and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number

36. The sums of first n terms of three arithmetic progressions are S1 S2 and S3 respectively. The first term of each A.P. is 1 and their common differences are 1, 2 and 3 respectively. Prove that S1 + S3 = 2S2.

37. The 17th term of an AP is 5 more than twice its 8th term. If the 11th term of the AP is 43, then find its nth term.

38. Find the 60th term of the AP 8, 10, 12, …, if it has a total of 60 terms and hence find the sum of its last 10 terms

39. Find the common difference of an A.P. whose first term is 5 and the sum of its first four terms is half the sum of the next four terms

40. The houses in a row are numbered conse cutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to sum of the numbers of houses following X.

1. 35

2. 20

3. Rs 160, Rs 140, Rs 120, Rs 100, Rs 80, Rs 60, and Rs 40.

4. 3rd, 10th, and nth terms are −1, −15, and 5 − 2n

5. 1661

6. 16, 8/3

7. 4 or 13

8. 10

9. −13, −8, and −3.

10. 60 multiples of 4 between 10 and 250.

11. 65th

12. 5th

13. No 

14. 35th and yes-8th

15. 40°, 60°, 80°

16. 28th

17. -

18. 8, 10,12,14

19. 15

20. 43

21. 100

22. 89

23. 16th term 111

24. 27

25. -925

26. n=16 d=8/3

27. 1,6,11……

28. 44550

29. 34th

30. 3,5,7….

31. -340

32. 151

33. 2, 10, 18, 26, 34….

34. -

35. 852

36. -

37. 4n-1

38. 1170

39. 2

40. 35

Solution link for Question 1 to Question 14 is https://byjus.com/maths/important-questions-class-10-maths-chapter-5-arithmetic-progressions/

Solution link for Question 1 to Question 14 is 40 is https://www.learncbse.in/important-questions-for-class-10-maths-chapter-5/

Machine Learning Notes

Introduction to machine learning: 

○ Definition

•   Machine learning is a type of artificial intelligence (AI), which means it's a computer program that can do things that usually require human intelligence.
•  Machine learning is a type of computer program that can learn from data and improve its performance over time.
• Machine learning is a field of computer science that enables machines to learn from data and improve their performance on a task without being explicitly programmed.
• It involves the development of algorithms and models that can learn patterns and insights from large datasets and make predictions or decisions based on that learning.
• machine learning is teaching machines to learn from experience, just like humans do, and use that learning to make better decisions or predictions.

○ Machine Learning & Human Learning. 

Machine Learning	Human Learning
Process	Algorithms and models are designed to learn patterns and insights from data	The brain processes and stores information through experience and practice
Input	Data is fed to the machine	Information is perceived through the senses
Output	Predictions, decisions, or classifications based on data analysis	Thoughts, actions, and behavior
Error	Error is minimized through optimization of algorithms	Error is minimized through practice and feedback
Generalization	Machine learning algorithms generalize from training data to new data	Humans generalize learning to new situations and contexts
Explanation	Machine learning models provide explanations for their predictions	Humans can provide explanations for their decisions and thought processes
Adaptation	Machine learning algorithms can be retrained to adapt to new data or tasks	Humans can learn new skills or adapt to changing environments

9

○ Scope of Machine Learning. 

○ Application of machine learning.