If an=2n+3, find S20
Answers
Answer:
S20=480
Step-by-step explanation:
hope this answer helped you.
Answer :
S(20) = 480
Note :
• Σc = c + c + c + . . . n times = nc , c is any constant .
• Σc·a(n) = c·Σa(n) , c is any constant .
• Σ[a(n) ± b(n)] = Σa(n) ± Σb(n)
• Sum of first n natural numbers is given as ; Σn = 1 + 2 + 3 + . . . + n = n(n+1)/2
• Sum of squares of n natural numbers is given as ;
Σn² = 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6
• Sum of cubes of n natural numbers is given as ;
Σn³ = 1³ + 2³ + 3³ + . . . + n³ = [n(n+1)/2]²
• S(n) = Σa(n)
Solution :
- Given : a(n) = 2n + 3
- To find : S(20) = ?
We have ;
a(n) = 2n + 3
Also ,
We know that , S(n) = Σa(n)
Thus ,
=> S(n) = Σa(n)
=> S(n) = Σ(2n + 3)
=> S(n) = Σ2n + Σ3
=> S(n) = 2Σn + Σ3
=> S(n) = 2[n(n+1)/2] + 3n
=> S(n) = n(n + 1) + 3n
=> S(n) = n(n + 1 + 3)
=> S(n) = n(n + 4)
Now ,
=> S(20) = 20×(20 + 4)
=> S(20) = 20×24
=> S(20) = 480
Hence , S(20) = 480 .
Alternative method :
Note :
★ A linear polynomial in variable n always represents the nth term of an AP .
★ A quadratic polynomial in variable n always represents the sum of n terms of an AP .
★ A.P. (Arithmetic Progression) : A sequence in which the difference between the consecutive terms are equal is said to be in A.P.
★ If a1 , a2 , a3 , . . . , an are in AP , then
a2 - a1 = a3 - a2 = a4 - a3 = . . .
★ The common difference of an AP is given by ; d = a(n) - a(n-1) .
★ The sum of nth terms of an AP is given by ; S(n) = (n/2)×[ 2a + (n - 1)d ] .
or S(n) = (n/2)×(a + l) , l is the last term .
Solution :
- Given : a(n) = 2n + 3
- To find : S(20) = ?
We have ;
a(n) = 2n + 3
Thus ,
a(1) = 2×1 + 3 = 2 + 3 = 5
a(2) = 2×2 + 3 = 4 + 3 = 7
Now ,
The common difference will be ;
d = a(2) - a(1) = 7 - 5 = 2
Now ,
We know that , the sum of first n terms of an AP is given as ;
S(n) = (n/2)•[2a + (n -1)d]
Thus ,
The sum of first 20 terms will be ;
=> S(20) = (20/2)•[2a + (20 - 1)d]
=> S(20) = 10•(2a + 19d)
=> S(20) = 10•(2•5 + 19•2)
=> S(20) = 10•(10 + 38)
=> S(20) = 10•48
=> S(20) = 480