Prove \( 2 \times 1! + 5 \times 2! + 10 \times 3! + \cdots + (n^2+1)n! = n(n+1)! \). Step 1 Show it is true for \( n=1 \). \( \begin{align} &\text{LHS} = (1^2+1) \times 1! = 2 \\ &\text{RHS} = 1 \times (1+1)! = 2 \\ &\text{LHS} = \text{RHS} \\ &\text{Therefore it is […]
Author Archives: iitutor
Determining Initial Values | Principles of Mathematical Induction
Prove \( 1+3+5+\cdots+(2n+1) = (n+1)^2 \). Step 1 Show it is true for \( n=0 \) by mathematical induction. \( \begin{align} &\text{LHS} = 2 \times 0 +1 = 1 \\ &\text{RHS} = (0+1)^2 = 1 \\ &\text{LHS} = \text{RHS} \\ &\text{Therefore, it is true for } n=0 \end{align} \) Step 2 Assume that it is […]
Evaluating Algebraic Expressions using Ratios
Given \( \displaystyle \frac{a+b}{3} = \frac{b+c}{4} = \frac{c+a}{5} \), evaluate the following. (a) \( a : b : c \) \( \displaystyle \begin{align} \require{AMSsymbols} \text{Let }\frac{a+b}{3} &= \frac{b+c}{4} = \frac{c+a}{5} = k \\ a+b &= 3k \cdots (1) \\ b+c &= 4k \cdots (2) \\ c+a &= 5k \cdots (3) \\ (a+b)-(c+a) &= 3k-5k &\color{green}{(1)-(3)} \\ […]
Divisibility Proofs by Algebraic Method
Example 1 (a) Prove the product of three consecutive integers is divisible by \( 3 \). \( \begin{align} &\text{Consider three consecutive positive integers are } n,n+1,n+2 \text{ where } n \text{ is a positive integer} \\ &\text{So the product of three consecutive positive integers: } n(n+1)(n+2) \\ &n \text{ can be in the form […]
Integration by Parts – An Ultimate Guide
Integration by parts is a technique for evaluating definite integrals or indefinite integrals. This process is often called Partial Integration and is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is based on the formula: $$ \large \displaystyle […]
Finding the Sum of Geometric Series of n Terms being Inserted between 1 and 2
Suppose \( n \) consecutive geometric terms are inserted between \( 1 \) and \( 2 \). Write the sum of these \( n \) terms in terms of \( n \). $$ \Large \underbrace{1, \overbrace{u_1, u_2, u_3, \cdots, u_{n}}^{S_n}, 2}_{S_{n+2}} $$ \( \require{AMSsymbols} \displaystyle \begin{align} S_{n+2} &= 1+u_1 + u_2 + u_3 + \cdots […]
Factorising Cubic Expressions with Rotating Three Variables such as \( a^2+a^2c+ac^2-b^3-b^2c-bc^2 \)
For these sorts of factorisations involving rotation of variables, it would be a good idea to rearrange regarding only a specific variable. Example 1 Factorise \( a^2+a^2c+ac^2-b^3-b^2c-bc^2 \). \( \begin{align} &= (ac^2-b^c)+(a^2c-b^2c)+(a^3-b^3) \\ &= (a-b)c^2+(a^2-b^2)c+(a^3-b^3) \\ &= (a-b)c^2+(a-b)(a+b)c+(a-b)(a^2+ab+b^2) \\ &= (a-b)\left[ c^2+(a+b)c+(a^2+ab+b^2) \right] \\ &= (a-b)(a^2+b^2+c^2+ab+bc+ca) \end{align} \) Example 2 Factorise \( x^3+3ax^2+(3a^2-b^2)x+a^3-ab^2 \). \( […]
Factorising Quartic Expressions with two Quadratic Factors and a Remainder, such as \( (x^2+3x-2)(x^2+3x+4)-27 \) \( (x^2-8x+12)(x^2-7x+12)-6x^2 \)
Example 1 Factorise \( (x^2+3x-2)(x^2+3x+4)-27 \). \( \require{AMSsymbols} \begin{align} &= \left[ (\bbox[yellow]{x^2+3x})-2 \right] \left[ (\bbox[yellow]{x^2+3x})+4 \right]-27 \\ &= (\bbox[yellow]{x^2+3x})^2 + 2(\bbox[yellow]{x^2+3x})-8-27 \\ &= (\bbox[yellow]{x^2+3x})^2 + 2(\bbox[yellow]{x^2+3x})-35 \\ &= (\bbox[yellow]{x^2+3x}+7)(\bbox[yellow]{x^2+3x}-5) \end{align} \) Example 2 Factorise \( (x^2-8x+12)(x^2-7x+12)-6x^2 \). \( \require{AMSsymbols} \begin{align} &= \left[ (x^2+12)-8x \right] \left[ (x^2+12)-7x \right]-6x^2 \\ &= (x^2+12)^2 -15x(x^2+12) + 56x^2-6x^2 \\ &= […]
Factorising Quartics with Four Factors and a Remainder such as \( (x-1)(x-3)(x+2)(x+4)+24 \)
Example 1 Factorise \( (x-1)(x-3)(x+2)(x+4)+24 \). \( \require{AMSsymbols} \begin{align} &= (x-1)(x+2) \times (x-3)(x+4) + 24 \\ &= (x^2+x-2) \times (x^2+x-12) +24 \\ &= \left[(x^2+x)-2\right] \times \left[(x^2+x)-12\right] +24 \\ &= (x^2+x)^2-14(x^2+x)+24+24 \\ &= (x^2+x)^2\bbox[aqua]{-14(x^2+x)}+48 \end{align} \) \( \require{AMSsymbols} \begin{array} {ccr} &\bbox[yellow]{x^2+x} &\bbox[pink,3px]{-6} &\bbox[pink]{-6(x^2+x)} \\ &\bbox[pink]{x^2+x} &\bbox[yellow,3px]{-8} &\bbox[yellow]{-8(x^2+x)} \\ \hline &&&\bbox[aqua]{-14(x^2+x)} \end{array} \) \( \require{AMSsymbols} \begin{align} &= […]
Factorising Quadratics with Six Terms of \( x \) and \( y \) such as \( x^2 + 2xy + 5x + 5y + y^2 + 6 \)
Sometimes students may encounter complex quadratics factorise involving \( x^2, y^2, xy, x \) and \( y \). Consider factorising by only either \( x \) or \( y \). The following examples take through factorising \( y \) first, then \( x\). Example 1 Factorise \( x^2 + 2xy + 5x + y^2+ 5y […]