the number of potteries produced.
(ii) How many terms of the A.P. 16, 14, 12, ... are needed to give the sum 60? Explain
why do we get two answers.
the following
Answers
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Step-by-step explanation:
Given:-
The terms of the A.P. 16, 14, 12, ...
To find:-
How many terms of the A.P. 16, 14, 12, ... are needed to give the sum 60?
Solution:-
Given that
16,14,12,... are in the AP
First term (a) = 16
Second term = 14
Common difference (d)=14-16 = -2
Given sum = 60
Let the number of terms are needed to give the sum 60 be "n"
Therefore, Sn = 60
We know that
The sum of the n terms of an AP is denoted by Sn and defined by Sn = (n/2)[2a+(n-1)d]
=>(n/2)[2a+(n-1)d] = 60
=>(n/2)[2×16 +(n-1)(-2)] = 60
=>(n/2)[32+(n-1)(-2)] = 60
=>(n/2)[32 -2n +2] = 60
=>(n/2)[34-2n] = 60
=>2(n/2)(17-n)=60
=>(n)(17-n) = 60
=>17n-n^2 = 60
=>17n-n^2-60 = 0
=>-n^2 +17n -60 = 0
=>n^2-17n +60 = 0
=>n^2-12n-5n +60 = 0
=>n(n-12)-5(n-12) = 0
=>(n-12)(n-5) = 0
=>n-12 = 0 or n-5 = 0
=>n=12 or n=5
Therefore, n = 12 or 5
Answer:-
The required number of terms are 12 or 5
Check:-
If the number of terms is 5 then the AP
16,14,12,10,8
Their sum = 16+14+12+10+8 =60
If the number of terms is 12 then the AP
16,14,12,10,8,6,4,2,0,-2,-4,-6
Their sum = 16+14+12+10+8+6+4+2+0-2-4-6
=>72-12 = 60
Verified the given relations
Used formulae:-
The sum of the n terms of an AP is denoted by Sn and defined by Sn = (n/2)[2a+(n-1)d]
Where,
a = First term
d = Common difference
n= number of terms
Sn = Sum of n terms in the AP
Clarification:-
We get a quadratic equation for the given problem ,
So a quadratic equations has at most two roots . That's why we get two answers for the number of terms and the two answers are satisfying the given AP for the given condition.