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26 Nov 2020, 04:41
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nth term of a certain sequence, for all n>2, can be obtained by subtracting from the previous term the term before that. If first term is 4 and second term is 8, what is the sum of first 100 terms of the sequence?
A.4
B.8
C.16
D.12
E.13
A.4
B.8
C.16
D.12
E.13
26 Nov 2020, 06:37
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indbos
nth term of a certain sequence, for all n>2, can be obtained by subtracting from the previous term the term before that. If first term is 4 and second term is 8, what is the sum of first 100 terms of the sequence?
A.4
B.8
C.16
D.12
E.13
A.4
B.8
C.16
D.12
E.13
When we have a series which is defined by a certain equation, then we see if the series repeats in a certain cycle.
n1 = 4
n2 = 8
n3 = 8 - 4 = 4
n4 = 4 - 8 = -4
n5 = -4 - 4 = -8
n6 = -8 - (-4) = -4
n7 = -4 - (-8) = 4
n8 = 4 - (-4) = 8
So every 6 terms will keep repeating in a cycle. The sum of every 6 terms = 4 + 8 + 4 + (-4) + (-8) + (-4) = 0
We want the syum of the first 100 terms. The largest number divisible by 6 is 96. Therefore the sum of the first 96 terms =
The sum of the last 4 terms = 4 + 8 + 4 - 4 = 12
Option D
Arun Kumar
General Discussion
15 Jun 2023, 07:15
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indbos
nth term of a certain sequence, for all n>2, can be obtained by subtracting from the previous term the term before that. If first term is 4 and second term is 8, what is the sum of first 100 terms of the sequence?
A.4
B.8
C.16
D.12
E.13
A.4
B.8
C.16
D.12
E.13
One of my students asked me to solve this. So here goes....
From the above solution, we know that:
Term1 = 4
Term2 = 8
Term3 = 4
Term4 = -4
Term5 = -8
Term6 = -4
Term7 = 4
Term8 = 8
Term9 = 4
Term10 = -4
.
.
.
We can see that the sequence has a cycle of 6.
In other words, the sequence looks like this: 4, 8, 4, -4, -8, -4, 4, 8, 4, -4, -8, -4, 4, 8, 4, -4, -8, -4, 4, 8, 4, -4, -8, -4, . . .
Now notice that 4 + 8 + 4 + (-4) + (-8) + (-4) = 0
This means each cycle of six terms has a sum of 0.
Since 96/6 = 16, we know that there are 16 complete cycles of 6 terms when we list the first 96 terms.
Since each cycle of 6 terms has a sum of 0, we can now conclude that the sum of the first 96 terms must be 0.
So now all we need to do is add 4 more terms.
That is, we need to add the first four terms of the cycle.
When we do so we get: 4 + 8 + 4 + (-4) = 12
Answer: D
