ESS Study Tips

❇️General math formulas ▎Arithmetic Sequence 1. nth Term of an Arithmetic Sequence: aₙ = a + (n-1)d where:    •  a  = first term    •  d  = common difference    •  n  = term number 2. Sum of the First n Terms of an Arithmetic Series: Sₙ = n / 2 (2a + (n-1)d) = n / 2 (a + l) where:    •  Sₙ  = sum of the first  n  terms    •  l  = last term ▎Geometric Sequence 3. nth Term of a Geometric Sequence: aₙ = arⁿ⁻¹ where:    •  a  = first term    •  r  = common ratio    •  n  = term number 4. Sum of the First n Terms of a Geometric Series: Sₙ = a 1 - rⁿ / 1 - r,   (r ≠ 1) 5. Sum of an Infinite Geometric Series: S = a / 1 - r,   (|r| < 1) ▎Fibonacci Sequence 6. Fibonacci Sequence:    The nth Fibonacci number can be defined recursively as: Fₙ = Fₙ₋₁ + Fₙ₋₂ with initial conditions  F₀ = 0, F₁ = 1 . ▎General Formulas 7. Factorial:    The factorial of a non-negative integer  n : n! = n × (n-1) × (n-2) × ... × 1 8. Binomial Theorem:    For any positive integer  n : (x + y)ⁿ = ∑ₖ₌₀ⁿ \binomnk xⁿ⁻ᵏ yᵏ

⚅sequences and series, questions with their answers: ▎Questions 1. What is the next term in the arithmetic sequence: 2, 5, 8, 11, ...?    a) 12     b) 13     c) 14     d) 15  2. Which of the following is a geometric series?    a) 1, 2, 4, 8, ...     b) 1, 3, 5, 7, ...     c) 1, 4, 9, 16, ...     d) 1, -1, 1, -1, ...  3. The sum of the first n terms of an arithmetic series is given by the formula:    a)  Sₙ = n/2 (a + l)      b)  Sₙ = n/2 (2a + (n-1)d)      c)  Sₙ = a + (n-1)d      d)  Sₙ = n ⋅ d   4. What is the common difference in the arithmetic sequence: 10, 7, 4, ...?    a) -3     b) -2     c) -1     d) 0  5. If the first term of a geometric sequence is 3 and the common ratio is 2, what is the fifth term?    a) 24     b) 48     c) 96     d) 192  6. What is the sum of the first five terms of the sequence: 1, 1/2, 1/4, ...?    a) 1.875     b) 2.0     c) 2.5     d) 3.0  7. In the Fibonacci sequence, what is the seventh term?    a) 8     b) 13     c) 21     d) 34  8. The nth term of an arithmetic sequence can be expressed as:    a)  aₙ = a + (n-1)d      b)  aₙ = a + nd      c)  aₙ = nd      d)  aₙ = n² + a   9. What is the sum of the infinite geometric series:  ⅓ + 1/9 + 1/27 + ... ?    a)  ½      b)  ¼      c)  ½ × ⅓      d)  ½ × ⅔   10. If the first term of an arithmetic series is 5 and the common difference is 3, what is the 10th term?     a) 32      b) 33      c) 34      d) 35  11. Which of the following sequences is neither arithmetic nor geometric?     a) 2, 4, 6, ...      b) 3, 9, 27, ...      c) 1, 2, 4, 8, ...      d) 1, -1, 0, -2, ...  12. The sum of the first n terms of a geometric series can be calculated using:     a)  Sₙ = a(1 - rⁿ)/(1 - r) , if  r ≠ 1       b)  Sₙ = arⁿ/(1 - r) , if  r < 1       c)  Sₙ = n(a + l)/2 , if it’s arithmetic      d) All of the above  13. What is the common ratio in the geometric sequence: 81, 27, 9, ...?     a)  ⅓       b)  ¼       c)  ½       d)  ¾ 14. If the sequence defined by  aₙ = n² + n + 1 , what is  a₄ ?     a) 17      b) 21      c) 25      d) 29  15. The nth term of the sequence defined by  aₙ = (-1)ⁿ n , what is  a₅ ?     a) -5      b) -4      c) -3      d) -2  16. A sequence where each term is half of the previous term is called:     a) Arithmetic sequence      b) Geometric sequence      c) Harmonic sequence      d) Quadratic sequence  17. What is the next term in the series: 5, 10, 20, ...?     a) 30      b) 40      c) 50      d) 60  18. If the sum of an infinite geometric series is  S = a/(1 - r) , what condition must hold for it to converge?     a)  r > -1       b)  |r| < 1       c)  r < 0       d)  r > 0 19. What is the sum of the first four terms of the arithmetic series: -2, -7, -12...?     a) -32      b) -36      c) -40      d) -44  20. In an arithmetic series where the first term is x and the common difference is y, what is the expression for the nth term?     a) x + (n-1)y      b) xy(n-1)      c) x(n+y-1)      d) x + ny  ▎Answers 1. b (13) 2. a (1, 2, 4, 8,...) 3. b ( Sₙ = n/2 (2a + (n-1)d)) 4. a (-3) 5. d (48) 6. b (2.0) 7. c (13) 8. a ( aₙ = a + (n-1)d) 9. a ( ½ ) 10. c (32) 11. d (1, -1, 0, -2,...) 12. a ( Sₙ = a(1 - rⁿ)/(1 - r), if r ≠ 1) 13. a ( ⅓ ) 14. b (21) 15. a (-5) 16. b (Geometric sequence) 17. b (40) 18. b ( |r| < 1) 19. b (-36) 20. a (x + (n-1)y) ≻───── ⋆✩⋆ ─────≺ https://t.me/ess_study_tips ≻───── ⋆✩⋆ ─────≺

⚅sequences and series, questions with their answers: ▎Questions 1. What is the next term in the arithmetic sequence: 2, 5, 8, 11, ...?    a) 12     b) 13     c) 14     d) 15  2. Which of the following is a geometric series?    a) 1, 2, 4, 8, ...     b) 1, 3, 5, 7, ...     c) 1, 4, 9, 16, ...     d) 1, -1, 1, -1, ...  3. The sum of the first n terms of an arithmetic series is given by the formula:    a)  Sₙ = n/2 (a + l)      b)  Sₙ = n/2 (2a + (n-1)d)      c)  Sₙ = a + (n-1)d      d)  Sₙ = n ⋅ d   4. What is the common difference in the arithmetic sequence: 10, 7, 4, ...?    a) -3     b) -2     c) -1     d) 0  5. If the first term of a geometric sequence is 3 and the common ratio is 2, what is the fifth term?    a) 24     b) 48     c) 96     d) 192  6. What is the sum of the first five terms of the sequence: 1, 1/2, 1/4, ...?    a) 1.875     b) 2.0     c) 2.5     d) 3.0  7. In the Fibonacci sequence, what is the seventh term?    a) 8     b) 13     c) 21     d) 34  8. The nth term of an arithmetic sequence can be expressed as:    a)  aₙ = a + (n-1)d      b)  aₙ = a + nd      c)  aₙ = nd      d)  aₙ = n² + a   9. What is the sum of the infinite geometric series:  ⅓ + 1/9 + 1/27 + ... ?    a)  ½      b)  ¼      c)  ½ × ⅓      d)  ½ × ⅔   10. If the first term of an arithmetic series is 5 and the common difference is 3, what is the 10th term?     a) 32      b) 33      c) 34      d) 35  11. Which of the following sequences is neither arithmetic nor geometric?     a) 2, 4, 6, ...      b) 3, 9, 27, ...      c) 1, 2, 4, 8, ...      d) 1, -1, 0, -2, ...  12. The sum of the first n terms of a geometric series can be calculated using:     a)  Sₙ = a(1 - rⁿ)/(1 - r) , if  r ≠ 1       b)  Sₙ = arⁿ/(1 - r) , if  r < 1       c)  Sₙ = n(a + l)/2 , if it’s arithmetic      d) All of the above  13. What is the common ratio in the geometric sequence: 81, 27, 9, ...?     a)  ⅓       b)  ¼       c)  ½       d)  ¾ 14. If the sequence defined by  aₙ = n² + n + 1 , what is  a₄ ?     a) 17      b) 21      c) 25      d) 29  15. The nth term of the sequence defined by  aₙ = (-1)ⁿ n , what is  a₅ ?     a) -5      b) -4      c) -3      d) -2  16. A sequence where each term is half of the previous term is called:     a) Arithmetic sequence      b) Geometric sequence      c) Harmonic sequence      d) Quadratic sequence  17. What is the next term in the series: 5, 10, 20, ...?     a) 30      b) 40      c) 50      d) 60  18. If the sum of an infinite geometric series is  S = a/(1 - r) , what condition must hold for it to converge?     a)  r > -1       b)  |r| < 1       c)  r < 0       d)  r > 0 19. What is the sum of the first four terms of the arithmetic series: -2, -7, -12...?     a) -32      b) -36      c) -40      d) -44  20. In an arithmetic series where the first term is x and the common difference is y, what is the expression for the nth term?     a) x + (n-1)y      b) xy(n-1)      c) x(n+y-1)      d) x + ny  ▎Answers 1. b (13) 2. a (1, 2, 4, 8,...) 3. b ( Sₙ = n/2 (2a + (n-1)d)) 4. a (-3) 5. d (48) 6. b (2.0) 7. c (13) 8. a ( aₙ = a + (n-1)d) 9. a ( ½ ) 10. c (32) 11. d (1, -1, 0, -2,...) 12. a ( Sₙ = a(1 - rⁿ)/(1 - r), if r ≠ 1) 13. a ( ⅓ ) 14. b (21) 15. a (-5) 16. b (Geometric sequence) 17. b (40) 18. b ( |r| < 1) 19. b (-36) 20. a (x + (n-1)y)

common physics choice questions about motion with answers. Question 1: A car travels 100 kilometers in 2 hours. What is its average speed? A) 200 km/h B) 50 km/h C) 100 km/h D) 25 km/h Answer: B) 50 km/h Explanation: Average speed is calculated by dividing the total distance traveled by the total time taken. Question 2: A ball is thrown straight up in the air. At its highest point, what is its velocity? A) 0 m/s B) 9.8 m/s C) -9.8 m/s D) Cannot be determined without more information Answer: A) 0 m/s Explanation: At the highest point, the ball momentarily stops before falling back down. Velocity is a vector quantity that includes both speed and direction, so the velocity is 0 m/s. Question 3: A runner completes a 400-meter race in 50 seconds. What is the runner's average speed? A) 8 m/s B) 10 m/s C) 12 m/s D) 16 m/s Answer: A) 8 m/s Explanation: Average speed = Total distance / Total time. 400 meters / 50 seconds = 8 m/s. Question 4: A car accelerates from rest to a speed of 20 m/s in 5 seconds. What is its acceleration? A) 2 m/s² B) 4 m/s² C) 6 m/s² D) 8 m/s² Answer: B) 4 m/s² Explanation: Acceleration is the change in velocity over time. Acceleration = (Final velocity - Initial velocity) / time = (20 m/s - 0 m/s) / 5 s = 4 m/s². Question 5: A ball is dropped from a height of 10 meters. Ignoring air resistance, what is the ball's velocity just before it hits the ground? A) 9.8 m/s B) 14 m/s C) 19.6 m/s D) Cannot be determined without more information Answer: b) 14 m/s Explanation: The acceleration due to gravity is 9.8 m/s². Using the equation: v² = u² + 2as, where: • v = final velocity • u = initial velocity (0 m/s) • a = acceleration due to gravity (9.8 m/s²) • s = distance (10 m) We get: v² = 0² + 2  9.8  10 = 196. Therefore, v = √196 = 14 m/s. Kindly share for your friends 🙏

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✅abbreviations used in history:✅ 31. BC: Before Christ 32. AD: Anno Domini (in the year of our Lord) 33. CE: Common Era 34. BCE: Before Common Era 35. WWI: World War I 36. WWII: World War II 37. UN: United Nations 38. EU: European Union 39. USA: United States of America 40. USSR: Union of Soviet Socialist Republics 41. NAACP: National Association for the Advancement of Colored People 42. NAFTA: North American Free Trade Agreement 43. NATO: North Atlantic Treaty Organization 44. UNESCO: United Nations Educational, Scientific and Cultural Organization 45. CIA: Central Intelligence Agency 46. FBI: Federal Bureau of Investigation 47. KGB: Komitet Gosudarstvennoy Bezopasnosti (Committee for State Security, Soviet Union) 48. NASA: National Aeronautics and Space Administration 49. GDP: Gross Domestic Product 50. FDR: Franklin D. Roosevelt https://t.me/jfuHCGff1sfgjjdsss/52

✅ abbreviations used in geography:✅ 26. N: North 27. S: South 28. E: East 29. W: West 30. Lat.: Latitude 31. Long.: Longitude 32. Alt.: Altitude 33. Ave.: Avenue 34. Blvd.: Boulevard 35. Cir.: Circle 36. Dr.: Drive 37. Rd.: Road 38. St.: Street 39. Hwy.: Highway 40. Co.: County 41. Pop.: Population 42. Temp.: Temperature 43. Precip.: Precipitation 44. Env.: Environment 45. GIS: Geographic Information System 46. DEM: Digital Elevation Model 47. GNP: Gross National Product 48. CBD: Central Business District 49. ECO: Ecological 50. WMO: World Meteorological Organization

✅ words short cuts ✅ mathematical abbreviations: 1. RHS: Right-hand side 2. LHS: Left-hand side 3. i.e.: Id est (Latin for "that is") 4. e.g.: Exempli gratia (Latin for "for example") 5. w.r.t.: With respect to 6. etc.: Et cetera (Latin for "and so forth") 7. s.t.: Such that 8. iff: If and only if 9. wlog: Without loss of generality 10. c.p.: Compare 11. resp.: Respectively 12. e.o.p.: End of proof 13. c.c.: Counterexample 14. w.l.o.g.: Without loss of generality 15. c.f.: Confer (Latin for "compare") 16. QED: Quod erat demonstrandum (Latin for "which was to be demonstrated") 17. DF: Definition 18. Thm: Theorem 19. Cor: Corollary no 20. Prop: Proposition 21. Lem: Lemma 22. Defn: Definition 23. Conj: Conjecture 24. Rem: Remark 25. Ex: Example 26. eqn.: Equation 27. sol.: Solution 28. seq.: Sequence 29. ser.: Series 30. lim: Limit 31. diff.: Differential 32. int.: Integral 33. det.: Determinant 34. vec.: Vector 35. mat.: Matrix 36. geom.: Geometry 37. alg.: Algebra 38. trig.: Trigonometry 39. calc.: Calculus 40. stat.: Statistics 41. prob.: Probability 42. num.: Number 43. comp.: Computation 44. graph.: Graph theory 45. lin.: Linear 46. quad.: Quadratic 47. geo.: Geometric 48. arith.: Arithmetic 49. poly.: Polynomial 50. func.: Function