a, a+d, a+2d, a+3d,.....,a+(n-1)d
Infinite GP
l=aₙ=a+(n-1)d
d=a₂-a₁=a₃-a₂=a₄-a₃
AM & Arithmetic
GM Sₙ=n/2[2a+(n-1)d]= n/2[a+l]= n/2[a+aₙ]
Progression
Relation
tₙ=l-(n-1)d
aₙ=Sₙ-S₍ₙ₋₁₎
If a,b,c are in GP then b=√(ac) & b is GM of a,c
Geometric #3 terms: a-d, a , a+d
Insert 'n' GMs btw a & b in GP a,G₁,G₂,G₃,.....Gₙ,b Mean Terms
#4 terms: a−3d,a−d,a+d,a+3d
R=(b/a)¹/⁽ⁿ⁺¹⁾ & G₁=aR; G₂=aR²; G₃=aR³; Gₙ=aRⁿ in AP
#5 terms: a−2d,a−d,a,a+d,a+2d
Sequence is GP if the ratio of Sequence & AM of a & b = (a+b)/2
consecutive term is same Arithmetic Mean
Series n AMs
d=(b-a)/(n+1)
algebraic pattern: a, ar, ar², ar³,.....arⁿ⁻¹ btw a & b
Aₙ=a+nd
r = common ratio = a₂/a₁ = a₃/a₂ = a₄/a₃
If a₁, a₂, a₃,.....aₙ are in AP then a₁±k, a₂±k,
l=aₙ=arⁿ⁻¹ a₃±k,.....aₙ±k are in AP with same common diff.
Geometric If a₁, a₂, a₃,.....aₙ are in AP then a₁k, a₂k, a₃k,.....aₙk
,r>1
Progression are in AP with common diff. multiplied/divided by k
Properties Sum of 1st & last terms is constant which is
,r<1
of AP equal to sum of equidistant terms from both ends
Sₙ=na ,r=1 If a,b,c.... are in AP then b=(a+c)/2
nᵗʰ term from end: tₙ=l(1/r)ⁿ⁻¹ The nᵗʰ term of AP is linear factor i.e. Aₙ+B &
sum of AP is quadratic expression: Sₙ=An²+Bn
Infinite GP
l=aₙ=a+(n-1)d
d=a₂-a₁=a₃-a₂=a₄-a₃
AM & Arithmetic
GM Sₙ=n/2[2a+(n-1)d]= n/2[a+l]= n/2[a+aₙ]
Progression
Relation
tₙ=l-(n-1)d
aₙ=Sₙ-S₍ₙ₋₁₎
If a,b,c are in GP then b=√(ac) & b is GM of a,c
Geometric #3 terms: a-d, a , a+d
Insert 'n' GMs btw a & b in GP a,G₁,G₂,G₃,.....Gₙ,b Mean Terms
#4 terms: a−3d,a−d,a+d,a+3d
R=(b/a)¹/⁽ⁿ⁺¹⁾ & G₁=aR; G₂=aR²; G₃=aR³; Gₙ=aRⁿ in AP
#5 terms: a−2d,a−d,a,a+d,a+2d
Sequence is GP if the ratio of Sequence & AM of a & b = (a+b)/2
consecutive term is same Arithmetic Mean
Series n AMs
d=(b-a)/(n+1)
algebraic pattern: a, ar, ar², ar³,.....arⁿ⁻¹ btw a & b
Aₙ=a+nd
r = common ratio = a₂/a₁ = a₃/a₂ = a₄/a₃
If a₁, a₂, a₃,.....aₙ are in AP then a₁±k, a₂±k,
l=aₙ=arⁿ⁻¹ a₃±k,.....aₙ±k are in AP with same common diff.
Geometric If a₁, a₂, a₃,.....aₙ are in AP then a₁k, a₂k, a₃k,.....aₙk
,r>1
Progression are in AP with common diff. multiplied/divided by k
Properties Sum of 1st & last terms is constant which is
,r<1
of AP equal to sum of equidistant terms from both ends
Sₙ=na ,r=1 If a,b,c.... are in AP then b=(a+c)/2
nᵗʰ term from end: tₙ=l(1/r)ⁿ⁻¹ The nᵗʰ term of AP is linear factor i.e. Aₙ+B &
sum of AP is quadratic expression: Sₙ=An²+Bn
