Maths formula
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1. This document lists various mathematical formulae related to algebra, quadratic equations, sequences and series, logarithms, and factorials. 2. Some key formulae include expressions for expanding (a + b)2, (a - b)2, and (a + b)3; the quadratic formula; formulas for the nth term and sum of first n terms of arithmetic and geometric progressions; and the factorial formula n! = n(n-1)! 3. Other formulas cover topics like logarithmic identities, properties of roots of quadratic equations, expanding binomial expressions, and evaluating sums involving powers of integers.
Maths formula
- 1. MATHEMATICAL FORMULAE Algebra 1. (a + b)2 = a2 + 2ab + b2 ; a2 + b2 = (a + b)2 − 2ab 2. (a − b)2 = a2 − 2ab + b2 ; a2 + b2 = (a − b)2 + 2ab 3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) 4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a + b)3 − 3ab(a + b) 5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a − b)3 + 3ab(a − b) 6. a2 − b2 = (a + b)(a − b) 7. a3 − b3 = (a − b)(a2 + ab + b2 ) 8. a3 + b3 = (a + b)(a2 − ab + b2 ) 9. an − bn = (a − b)(an−1 + an−2 b + an−3 b2 + · · · + bn−1 ) 10. an = a.a.a . . . n times 11. am .an = am+n 12. am an = am−n if m > n = 1 if m = n = 1 an−m if m < n; a ∈ R, a = 0 13. (am )n = amn = (an )m 14. (ab)n = an .bn 15. a b n = an bn 16. a0 = 1 where a ∈ R, a = 0 17. a−n = 1 an , an = 1 a−n 18. ap/q = q √ ap 19. If am = an and a = ±1, a = 0 then m = n 20. If an = bn where n = 0, then a = ±b 21. If √ x, √ y are quadratic surds and if a + √ x = √ y, then a = 0 and x = y 22. If √ x, √ y are quadratic surds and if a + √ x = b + √ y then a = b and x = y 23. If a, m, n are positive real numbers and a = 1, then loga mn = loga m+loga n 24. If a, m, n are positive real numbers, a = 1, then loga m n = loga m−loga n 25. If a and m are positive real numbers, a = 1 then loga mn = n loga m 26. If a, b and k are positive real numbers, b = 1, k = 1, then logb a = logk a logk b 27. logb a = 1 loga b where a, b are positive real numbers, a = 1, b = 1 28. if a, m, n are positive real numbers, a = 1 and if loga m = loga n, then m = n Typeset by AMS-TEX
- 2. 2 29. if a + ib = 0 where i = √ −1, then a = b = 0 30. if a + ib = x + iy, where i = √ −1, then a = x and b = y 31. The roots of the quadratic equation ax2 +bx+c = 0; a = 0 are −b ± √ b2 − 4ac 2a The solution set of the equation is −b + √ ∆ 2a , −b − √ ∆ 2a where ∆ = discriminant = b2 − 4ac 32. The roots are real and distinct if ∆ > 0. 33. The roots are real and coincident if ∆ = 0. 34. The roots are non-real if ∆ < 0. 35. If α and β are the roots of the equation ax2 + bx + c = 0, a = 0 then i) α + β = −b a = − coeff. of x coeff. of x2 ii) α · β = c a = constant term coeff. of x2 36. The quadratic equation whose roots are α and β is (x − α)(x − β) = 0 i.e. x2 − (α + β)x + αβ = 0 i.e. x2 − Sx + P = 0 where S =Sum of the roots and P =Product of the roots. 37. For an arithmetic progression (A.P.) whose first term is (a) and the common difference is (d). i) nth term= tn = a + (n − 1)d ii) The sum of the first (n) terms = Sn = n 2 (a + l) = n 2 {2a + (n − 1)d} where l =last term= a + (n − 1)d. 38. For a geometric progression (G.P.) whose first term is (a) and common ratio is (γ), i) nth term= tn = aγn−1 . ii) The sum of the first (n) terms: Sn = a(1 − γn ) 1 − γ ifγ < 1 = a(γn − 1) γ − 1 if γ > 1 = na if γ = 1 . 39. For any sequence {tn}, Sn − Sn−1 = tn where Sn =Sum of the first (n) terms. 40. n γ=1 γ = 1 + 2 + 3 + · · · + n = n 2 (n + 1). 41. n γ=1 γ2 = 12 + 22 + 32 + · · · + n2 = n 6 (n + 1)(2n + 1).
- 3. 3 42. n γ=1 γ3 = 13 + 23 + 33 + 43 + · · · + n3 = n2 4 (n + 1)2 . 43. n! = (1).(2).(3). . . . .(n − 1).n. 44. n! = n(n − 1)! = n(n − 1)(n − 2)! = . . . . . 45. 0! = 1. 46. (a + b)n = an + nan−1 b + n(n − 1) 2! an−2 b2 + n(n − 1)(n − 2) 3! an−3 b3 + · · · + bn , n > 1.