Progression | Maths

• Arithmetic progression (AP) is a sequence of numbers satisfying the condition that the difference between any two consecutive numbers is constant
• If the first term is denoted as ‘a’

The common difference as ‘d’

The number of terms as n.  

Then nth term is an = a + (n – 1)d.

• Sum of n terms,

Sn = n/2 [2a + (n - 1) d]

= n/2 (first term + last term).

• Sum of the first n natural numbers

1 + 2 + 3 + … + n = n(n+1)/2

• Sum of squares of the first n natural numbers

12+ 2²+ 3²+ …. + n² = n(n + 1)(2n + 1)/6

• Sum of cubes of first n natural numbers

1³+ 2³ + 3³ + …+ n³ = (1 + 2 + 3 + …+ n)² = [ n(n + 1)/2]²

• Sum of the first n odd numbers

1 + 3 + … + (2n – 1) = n²

• Sum of the first n even numbers

2+4+...+2n= n(n+1)

• If a,b,c are three consecutive terms of an AP then 2b=a+c
• Geometric progression (GP) is a sequence of  terms in which each succeeding term is obtained by multiplying its preceding term by a constant
• If first term is  denoted as ‘a’

The common ratio as ‘r’

Then nth term an = arn-1

• Sum of n terms

Sn = a(rn -1)/(r-1) where r>1

Sn = a(1-rn)/(1-r) where 1>r

• If a,b,c are three consecutive terms of an GP then b²=ac

Questions
1.20-ൽ താഴെയുള്ള ഉള്ള ഒറ്റ സംഖ്യകളുടെ തുകയെത്ര    ?  (LD Clerk SR for ST only 2020)
1+3+5+... +19
n ഒറ്റ സംഖ്യകളുടെ തുക (Sum of the first n odd numbers) = n²
ഇവിടെ 1 -19 വരെ 10 ഒറ്റ സംഖ്യകൾ
10² =100

2018
1.If 13, X, 22 are three consecutive terms of an AP. What is the value of X ? (Women Civil Excise Officer 2018)
2X=22+13
2X=35
X=17.5

2.n²,2n²,3n²,... ഒരു സമാന്തര ശ്രേണിയിലെ തുടർച്ചയായ പദങ്ങളായൽ ശ്രേണിയുടെ പൊതുവ്യത്യാസം എത്ര? (STORE KEEPER-KSFDC 2018)
2n²-n²= n²

3.If the nth term of an AP is 2n+1. Then the sum of the first three terms is (Beat Forest Officer 2018)
1st term = 2*1+1= 3
2nd term =2*2+1=5
3rd term = 2*3+1=7
Sum=3+5+7=15

4.62,55,48.... എന്ന ശ്രേണിയിലെ പത്താമത്തെ പദം ഏത്? (WOMEN POLICE CONSTABLE NCA 2018)
d=-7
nth term = a + (n – 1)d
10th term= 62-(9x7)=-1


5.1, 2, 4, 8, 16, ........ ഈ ശ്രേണിയിലെ 10 ആം സംഖ്യ ഏത് (AYAH 2018)
nth term an = arn-1
10 th term = 1 x 29 = 512

6.Which term of the sequence √3, 3, 3√3, ______ is 729 ? (Civil Police Officer 2018)
nth term an = arn-1

729 =  √3 x √3n-1  

36 =   √3n

√312 = √3n

n = 12

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