Mathematics Dictionary Dr. K. G. Shih

---

Examples in Algebra

Questions

Symbol Defintion Sqr(x) = Square root of x

Q01 | - Sequece : 1, 3, 7, 13, 21, .... Q02 | - Describe the curve y = (x^2)/((x^2 - 1)^2) Q03 | - Q04 | - Q05 | - Q06 | - Q07 | - Q08 | - Q09 | - Q10 | -

Answers

Q01. Sequence : 1, 3, 7, 13, 21, ... Question 1. Find next number 2. Find nth term T(n) 3. Find sum S(n) Solution Assume T(1) = 1 and T(2) = 3 Sequence : 1, 3, 7, 13, 21, ... 1st Diff : 2, 4, 6, 8, ........ 2nd Diff : 2, 2, 2, 2, ........ Answer 1. Next number is 6th number = 21 + 10 = 31 2. Find T(n) Since 2nd diff is common difference = 2 Hence T(n) = n^2 + b*n + c T(1) = 1 we have 1 = 1^2 + b + c or b + c = 0 T(2) = 3 we have 3 = 2^2 + 2*b + c or 2*b + c = -1 Solve above equations, we have b = -1 and c = 1 Hence T(n) = n^2 - n + 1 3. Find S(n) S(n) = Sum[n^2] - Sum[n] + Sum[1] S(n) = n*(n+1)*(2*n+1)/6 - n(n+1)/2 + n S(n) = n*((n+1)*(2*n+1) - 3*(n+1) + 6)/6 S(n) = n*(2*(n^2) + 4)/6 Verify Question 1 : T(6) = 6^2 - 6 + 1 = 31 Verify Question 3 S(3) = 1 + 3 + 7 = 11 S(3) = 3*(2*(3^2) + 4)/6 = 66/6 = 11 Go to Begin Q02. Describe the curve y = (x^2)/((x^2 - 1)^2) Asymptotes x = -1 y goes to infinite x = +1 y goes to infinite y-intercept : x = 0 y = 0 Signs of y x LT -1, y is negative x EQ -1, y is infinite, Hence curve from -0 to -infinite x GE -1 and x LT 0, y is negative, curve from -infinte to 0 (x = 0) x GT +0 and x LT 1, y is positive, curve from 0 to infinite x GT 1, y is positive, curve from infinite to +0 Concavity x LT -1, it must be concave downword x between -1 and 0, it must be concave downword x between 0 and 1, it must be concave upword x GT 1, it must be concave upword Go to Begin Q03. Answer Go to Begin Q04. Answer Go to Begin Q05. Answer Go to Begin Q06. Answer Go to Begin Q07. Answer Go to Begin Q08. Answer Go to Begin Q09. Answer Go to Begin Q10. Answer Go to Begin

---

Show Room of MD2002

Contact Dr. Shih

Math Examples Room

---

Copyright © Dr. K. G. Shih, Nova Scotia, Canada.

---