The
sum of first of n term of an
AP in 5n² - 3n, then find the common
difference
Answers
Step-by-step explanation:
Let an denotes the nth terms and Sn denotes the sum of the first n terms of given AP. Hence, the nth term ( an) = 10n +2 & the common Difference is 10.
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Answer :
Common difference , d = 10
Note :
★ 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 nth term of an AP is given by ;
a(n) = a + (n - 1)d .
★ If a , b , c are in AP , then 2b = a + c .
★ 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 .
★ The nth term of an AP can be also given by ; a(n) = S(n) - S(n-1) .
Solution :
- Given : S(n) = 5n² - 3n
- To find : d = ?
We know that ,
The nth term of an AP is given as ;
=> a(n) = S(n) - S(n-1)
=> a(n) = [ 5n² - 3n ] - [ 5(n - 1)² - 3(n - 1) ]
=> a(n) = 5n² - 3n - 5(n - 1)² + 3(n - 1)
=> a(n) = 5[ n² - (n - 1)² ] + 3( - n + n - 1 )
=> a(n) = 5[ n² - (n² - 2n + 1) ] + 3(-1)
=> a(n) = 5(n² - n² + 2n - 1) - 3
=> a(n) = 5(2n - 1) - 3
=> a(n) = 10n - 5 - 3
=> a(n) = 10n - 8
Also ,
We know that ,
The common difference is given by ;
=> d = a(n) - a(n - 1)
=> d = [ 10n - 8 ] - [ 10(n - 1) - 8 ]
=> d = 10n - 8 - 10(n - 1) + 8
=> d = 10n - 10(n - 1)
=> d = 10n - 10n + 10
=> d = 10
Hence ,
Common difference , d = 10 .
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Alternative method :
- Given : S(n) = 5n² - 3n
- To find : d = ?
Now ,
We know that ,
=> S(1) = a(1)
=> 5×1² - 3×1 = a(1)
=> 5 - 3 = a(1)
=> 2 = a(1)
=> a(1) = 2
Also ,
=> S(2) = a(1) + a(2)
=> 5×2² - 3×2 = 2 + a(2)
=> 20 - 6 = 2 + a(2)
=> a(2) = 20 - 6 - 2
=> a(2) = 20 - 8
=> a(2) = 12
Now ,
The common difference will be ;
=> d = a(2) - a(1)
=> d = 12 - 2
=> d = 10