source: ncert
topic: algebra
class_level: class_11
chapter: ch9
difficulty: intermediate
Class 11 | Ch9: Sequences and Series ARITHMETIC PROGRESSION: a, a+d, a+2d,... a_n=a+(n-1)d. S_n=n/2[2a+(n-1)d]=n/2[a+l]. Arithmetic Mean AM=(a+b)/2. AM between a and b. GEOMETRIC PROGRESSION: a, ar, ar^2,... a_n=ar^(n-1). S_n=a(r^n-1)/(r-1) for r≠1. S_n=na for r=1. Sum of infinite GP: S_inf=a/(1-r) for |r|<1. Geometric Mean GM=sqrt(ab). GM between a and b. RELATIONSHIP: AM>=GM (equality when a=b). HARMONIC PROGRESSION: reciprocals form AP. HM=2ab/(a+b). AM>=GM>=HM (with equality when a=b). SPECIAL SERIES: Sum of n natural numbers: n(n+1)/2. Sum of squares: n(n+1)(2n+1)/6. Sum of cubes: [n(n+1)/2]^2. SOLVED EXAMPLES: Example 1: GP 1+3+9+...+2187. a=1, r=3. 2187=3^7, so n=8. S8=1×(3^8-1)/(3-1)=(6561-1)/2=3280. Example 2: AM=10, GM=8. Find numbers. (a+b)/2=10 → a+b=20. sqrt(ab)=8 → ab=64. a and b are roots of x^2-20x+64=0 → (x-16)(x-4)=0. Numbers: 4 and 16. Example 3: Sum(2k+1) for k=1 to 20. =sum(2k)+sum(1)=2×sum(k)+20=2×(20×21/2)+20=420+20=440. Example 4: Sum 1/2+1/4+1/8+... (infinite GP). a=1/2, r=1/2. S=a/(1-r)=(1/2)/(1/2)=1. COMMON MISTAKES: Infinite GP sum only valid when |r|<1. AM>=GM: equality holds only when both numbers are equal. Sum of cubes=[sum of natural numbers]^2.