The sum of the s first 11 terms of an arithmetic sequence is 506 and the sum of its first 12 terms
is 600.
a) What is the 12th term of this sequence ?
Answers
Step-by-step explanation:
Given :-
The sum of the first 11 terms of an arithmetic sequence is 506 and the sum of its first 12 terms is 600.
To find :-
What is the 12th term of this sequence ?
Solution :-
Let the first term of an AP be a
Let the common difference be d
We know that
Sum of the first n terms of an AP
Sn = (n/2)[2a+(n-1)d]
Given that
The sum of the first 11 terms of an AP = 506
=> S11 = (11/2)[2a+(11-1)d] = 506
=> (11/2)[2a+10d] = 506
=> (11/2)×2(a+5d) = 506
=> 11(a+5d) = 506
=> a+5d = 506/11
=> a+5d = 46 ---------------(1)
On multiplying (1) with 2 then
=> 2a+10d = 92 -------------(2)
Given that
The sum of the first 12 terms of an AP = 600
=> S12= (12/2)[2a+(12-1)d] = 600
=> (12/2)[2a+11d] = 600
=> 6×(2a+11d) = 600
=> 2a+11d =600/6
=> 2a+11d = 100 ---------------(3)
On subtracting (2) from (3)
2a+11d = 100
2a+10d = 92
(-)
___________
0 + d = 8
___________
=> d = 8
Common difference = 8
On substituting the value of d in (1) then
=> a+5(8) = 46
=> a + 40 = 46
=> a = 46-40
=> a = 6
The first term = 6
We know that
nth term of an AP = an = a+(n-1)d
=> 12th term = a 12
=> a 12 = 6+(12-1)(8)
=> a 12 = 6+11(8)
=> a 12 = 6+88
=> a 12 = 94
Short cut:-
12th term = S 12 - S 11
=> a 12 = 600 - 506
=> a 12 = 94
Answer:-
The 12th term of the AP is 94
Used formulae:-
→ nth term of an AP = an = a+(n-1)d
→ Sum of the first n terms of an AP
Sn = (n/2)[2a+(n-1)d]
- a = First term
- d = Common difference
- n = Number of terms
Answer:
Step-by-step explanation:
