Advanced Sequences, Inequalities, and Straight Lines

Advanced Sequence Differences

• First Difference in Arithmetic Progression (AP):     * If the first difference of a sequence is in AP, the general term $t_n$ takes the quadratic form:     * $t_n = an^2 + bn + c$

• First Difference in Geometric Progression (GP):     * If the first difference is in GP, the general term takes the form:     * $t_n = a(r^n) + bn + c$

• Second Difference in AP:     * If the second difference of a sequence is in AP, the general term is a cubic polynomial:     * $t_n = an^3 + bn^2 + cn + d$

• Second Difference in GP:     * If the second difference is in GP, the general term takes the form:     * $t_n = a(r^n) + cn^2 + dn + e$

• Case Study Example:     * Sequence: $5 + 7 + 13 + 31 + 85 + \dots$ up to 10 terms. This was analyzed in the context of identifying the order of difference.

Telescopic Method of Difference

• Core Principle:     * Express the general term $t_r$ as a difference of two terms of a function, such as $t_n = f(n+1) - f(n)$.     * Summation formula: $S_n = \sum_{r=1}^n t_r = [f(2) - f(1)] + [f(3) - f(2)] + \dots + [f(n+1) - f(n)]$     * Cancellation occurs, leaving only the first and last terms: $S_n = f(n+1) - f(1)$.

• Application Example 1 (Product of Odds):     * Find the sum of 10 terms of $S = (1 \times 3 \times 5) + (3 \times 5 \times 7) + (5 \times 7 \times 9) + \dots$     * General term: $t_r = (2r - 1)(2r + 1)(2r + 3)$.     * Transformation: Multiply and divide by the constant difference of the extended sequence. Use $(2r + 5) - (2r - 3) = 8$.     * $t_r = \frac{1}{8} [(2r-1)(2r+1)(2r+3)(2r+5) - (2r-3)(2r-1)(2r+1)(2r+3)]$     * Final Sum $S_{10} = \frac{1}{8} [f(10) - f(0)] = \frac{1}{8} [(19 \times 21 \times 23 \times 25) + 15]$.

• Application Example 2 (Fractions with Odds):     * Series: $S_n = \frac{1}{1 \times 3} + \frac{2}{1 \times 3 \times 5} + \frac{3}{1 \times 3 \times 5 \times 7} + \dots$     * General term: $t_r = \frac{r}{1 \times 3 \times 5 \dots (2r+1)}$.     * Transformation: Multiply by 2. $2r = (2r + 1) - 1$.     * $t_r = \frac{1}{2} [\frac{(2r+1) - 1}{1 \times 3 \dots (2r+1)}] = \frac{1}{2} [\frac{1}{1 \times 3 \dots (2r-1)} - \frac{1}{1 \times 3 \dots (2r+1)}]$.     * Sum $S_n = \frac{1}{2} [1 - \frac{1}{1 \times 3 \dots (2n+1)}]$.

• JEE Mains 2021 Question:     * Calculate $\frac{1}{3^2-1} + \frac{1}{5^2-1} + \frac{1}{7^2-1} + \dots + \frac{1}{201^2-1}$.     * $t_r = \frac{1}{(2r+1)^2 - 1} = \frac{1}{4r(r+1)} = \frac{1}{4} [\frac{1}{r} - \frac{1}{r+1}]$.     * Sum for $n=100$: $\frac{1}{4} [1 - \frac{1}{101}] = \frac{25}{101}$.

• JEE Mains 2021 Question 2:     * Sum of 10 terms of $\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + \dots$     * General term: $t_r = \frac{2r+1}{r^2(r+1)^2} = \frac{(r+1)^2 - r^2}{r^2(r+1)^2} = \frac{1}{r^2} - \frac{1}{(r+1)^2}$.     * Result: $1 - \frac{1}{11^2} = \frac{120}{121}$.

Factorial Series and Summation

• Product Property:     * $n \times n! = ((n+1) - 1)n! = (n+1)! - n!$

• Example Case:     * $S = 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \dots + 100 \cdot 100!$     * $t_r = (r+1)! - r!$     * $S = 101! - 1!$

• Example Case (Fractional Factorials):     * $S = \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \dots + \frac{50}{51!}$     * $t_r = \frac{r}{(r+1)!} = \frac{(r+1) - 1}{(r+1)!} = \frac{1}{r!} - \frac{1}{(r+1)!}$.     * Sum: $1 - \frac{1}{51!}$.

Arithmetic, Geometric, Harmonic, and Root Mean Square Inequalities

• Theorem for Positive Real Numbers:     * For positive numbers $a_1, a_2, \dots, a_n$, the following hierarchy holds:     * $RMS \ge AM \ge GM \ge HM$

• Definitions for Two Positive Numbers ($a, b$):     * Arithmetic Mean (AM): $\frac{a+b}{2}$     * Geometric Mean (GM): $\sqrt{ab}$     * Harmonic Mean (HM): $\frac{2ab}{a+b}$     * Root Mean Square (RMS): $\sqrt{\frac{a^2+b^2}{2}}$

• General Formulas for $n$ Numbers:     * $AM = \frac{\sum a_i}{n}$     * $GM = (a_1 a_2 \dots a_n)^{\frac{1}{n}}$     * $HM = \frac{n}{\sum \frac{1}{a_i}}$     * $RMS = \sqrt{\frac{\sum a_i^2}{n}}$

• Equality Condition:     * The equality $AM = GM = HM = RMS$ holds if and only if $a_1 = a_2 = \dots = a_n$.

• Application - Minimum Values:     * For $f(x) = x + \frac{1}{x}$ where x > 0, since $AM \ge GM$: $\frac{x + \frac{1}{x}}{2} \ge \sqrt{x \cdot \frac{1}{x}} = 1\implies x + \frac{1}{x} \ge 2$. Range: $[2, \infty)$.     * JEE Advanced 2011: Minimum value of $a^{-5} + a^{-4} + 3a^{-3} + 1 + a^8 + a^{10}$ for a > 0.         * Split $3a^{-3}$ into $a^{-3} + a^{-3} + a^{-3}$. There are 8 terms total.         * Check product: $(a^{-5})(a^{-4})(a^{-3})^3(1)(a^8)(a^{10}) = a^{-5-4-9+18} = a^0 = 1$.         * $AM \ge GM \implies \frac{\text{Sum}}{8} \ge (1)^{\frac{1}{8}} \implies \text{Minimum Sum} = 8$.

Power Series Expansions and Maclaurin Series

• Maclaurin Series Formula:     * $f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots$

• Exponential Series ($e^x$):     * $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{r=0}^{\infty} \frac{x^r}{r!}$     * $e^{-x} = 1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots$

• Combined Exponential Sums:     * $\frac{e^x + e^{-x}}{2} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots$     * $\frac{e^x - e^{-x}}{2} = \frac{x}{1!} + \frac{x^3}{3!} + \frac{x^5}{5!} + \dots$

• Logarithmic Series:     * Valid for -1 < x \le 1: $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$     * $\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots$

• Homework Exercise:     * Evaluate $\sum_{n=0}^{\infty} \frac{n^3}{n!}$.

Straight Lines - Fundamentals of Coordinate Geometry

• Distances from Axes:     * Distance of point $(x, y)$ from X-axis = $|y|$ (Modulus of Ordinate).     * Distance of point $(x, y)$ from Y-axis = $|x|$ (Modulus of Abscissa).

• Relative Movement:     * Moving Right/Left: y-coordinate reflects no change; x-coordinate increases/decreases.     * Moving Up/Down: x-coordinate reflects no change; y-coordinate increases/decreases.

• Quadrant Logic (JEE Advanced 2011 Example):     * Point coordinates involving logarithms: Determining signs of terms like $5 + \log_2(x)$ to identify the quadrant.

• Section Formula:     * Internal Division: $P(x,y) = (\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n})$.     * External Division: Replace $n$ with $-n$.     * Harmonic Conjugates: If $P$ divides $AB$ internally and $Q$ divides $AB$ externally in ratio $m:n$, then $P$ and $Q$ are harmonic conjugates. Distances $AP, AB, AQ$ satisfy $\frac{2}{AB} = \frac{1}{AP} + \frac{1}{AQ}$.

• Collinearity of Three Points:     * Conditions: Area of triangle is 0; Slope of $AB$ = Slope of $BC$; Section formula is applicable with a consistent ratio ($\lambda$).

Centers of Triangles

• Centroid ($G$):     * Intersection of medians. Divides each median in ratio $2:1$ from the vertex.     * Coordinates: $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$.     * Area Property: $G$ divides a triangle into 3 triangles of equal area and 6 smaller triangles of equal area.

• Incenter ($I$):     * Intersection of internal angle bisectors. Center of the circle touching all sides.     * Equidistant from all sides.     * Coordinates: $(\frac{ax_1+bx_2+cx_3}{a+b+c}, \frac{ay_1+by_2+cy_3}{a+b+c})$.

• Circumcenter ($O$):     * Intersection of perpendicular bisectors. Center of circle passing through vertices.     * For a Right Triangle: Midpoint of the hypotenuse.

• Orthocenter ($H$):     * Intersection of altitudes.     * For a Right Triangle: The vertex where the $90^{\circ}$ angle is formed.

• The Euler Line ($O-G-H$ Rule):     * In any non-equilateral triangle, the Orthocenter ($H$), Centroid ($G$), and Circumcenter ($O$) are collinear.     * Centroid ($G$) divides the segment $HO$ in the ratio $2:1$.     * O-N-G-C mnemonic: Orthocenter ($H$), Nine-point center ($N$), Centroid ($G$), Circumcenter ($O$).

• Special Triangle Properties:     * Equilateral: All centers coincide ($H = G = I = O$).     * Isosceles: All centers are collinear.     * Image Property: The image of the Orthocenter ($H$) with respect to any side lies on the circumcircle.

Slope and Intercepts

• Slope ($m$):     * $m = \tan(\theta)$ where $\theta$ is the angle with the positive X-axis.     * Two-point form: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Parallel and Perpendicular Lines:     * Parallel: $m_1 = m_2$.     * Perpendicular: $m_1 \times m_2 = -1$.

• Angle Between Two Lines:     * $\tan(\theta) = |\frac{m_1 - m_2}{1 + m_1 m_2}|$

• Line Intercept Properties:     * Equal Intercepts: Slopes $m = -1$.     * Equal Length of Intercepts: Slope $m = \pm 1$.     * Equal Magnitude but Opposite Sign: Slope $m = 1$.     * Equally Inclined with Axes: Slope $m = \pm 1$.

Questions & Discussion

• Question: How many lines can pass through one point?

• Response: Infinite lines. If you have both a point and a fixed slope, then exactly one line is defined.

• Question: What happens to the slope of a vertical line?

• Response: The angle is $90^\circ$, so $\tan(90^\circ)$ is undefined. Thus, the slope is not defined.

• Question: In a right-angled triangle, where is the circumcenter located?

• Response: It is exactly at the midpoint of the hypotenuse.