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The sequence of positive numbers s1, S2, S3 … Sn … is define
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24 Jan 2020, 06:32
24 Jan 2020, 06:32
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The sequence of positive numbers \(S_1, S_2, S_3 … S_n …\) is defined by \(S_n = S_{n−1} + 5\) for \(n ≥ 2\). If \(S_1 = 7\), then which of the following is an expression for the nth term in the sequence?
A.5n − 5
B. 5n − 2
C. 5n
D. 5n + 2
E. 5n + 7
Kudos for the right answer and explanation
A.5n − 5
B. 5n − 2
C. 5n
D. 5n + 2
E. 5n + 7
Kudos for the right answer and explanation
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Re: The sequence of positive numbers s1, S2, S3 … Sn … is define
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24 Jan 2020, 13:22
24 Jan 2020, 13:22
1
Solution:
\((S_n = S_{n−1} + 5)\) for \(n ≥ 2\)
Also, we have: \(S_1 = 7\)
Thus, for n = 2: \(S_2 = S_1 + 5\) = 12
In the options, let us plugin n = 2 and determine which options gives us 12:
Option A: (5n - 5) becomes 5 at n = 2 --- incorrect
Option B: (5n - 2) becomes 8 at n = 2 --- incorrect
Option C: (5n) becomes 10 at n = 2 --- incorrect
Option D: (5n + 2) becomes 12 at n = 2 --- correct
Option E: (5n + 7) becomes 17 at n = 2 --- incorrect
Answer D
Alternate approach:
We know that: \(S_n = S_{n-1} + 5\)
This implies that each term is 5 greater than the previous term
Thus, the series is an Arithmetic Progression (AP)
The nth term of an AP having first term as \(a\) and difference between consecutive terms as \(d\) is \(a + (n - 1)d\)
Thus, in our example: \(a = 7\) and \(d = 5\)
=> The nth term = 7 + (n - 1) * 5 = 5n + 2
Answer D
\((S_n = S_{n−1} + 5)\) for \(n ≥ 2\)
Also, we have: \(S_1 = 7\)
Thus, for n = 2: \(S_2 = S_1 + 5\) = 12
In the options, let us plugin n = 2 and determine which options gives us 12:
Option A: (5n - 5) becomes 5 at n = 2 --- incorrect
Option B: (5n - 2) becomes 8 at n = 2 --- incorrect
Option C: (5n) becomes 10 at n = 2 --- incorrect
Option D: (5n + 2) becomes 12 at n = 2 --- correct
Option E: (5n + 7) becomes 17 at n = 2 --- incorrect
Answer D
Alternate approach:
We know that: \(S_n = S_{n-1} + 5\)
This implies that each term is 5 greater than the previous term
Thus, the series is an Arithmetic Progression (AP)
The nth term of an AP having first term as \(a\) and difference between consecutive terms as \(d\) is \(a + (n - 1)d\)
Thus, in our example: \(a = 7\) and \(d = 5\)
=> The nth term = 7 + (n - 1) * 5 = 5n + 2
Answer D
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Re: The sequence of positive numbers s1, S2, S3 … Sn … is define
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25 Jan 2020, 11:30
25 Jan 2020, 11:30
1
Carcass wrote:
The sequence of positive numbers \(S_1, S_2, S_3 … S_n …\) is defined by \(S_n = S_{n−1} + 5\) for \(n ≥ 2\). If \(S_1 = 7\), then which of the following is an expression for the nth term in the sequence?
A.5n − 5
B. 5n − 2
C. 5n
D. 5n + 2
E. 5n + 7
Kudos for the right answer and explanation
A.5n − 5
B. 5n − 2
C. 5n
D. 5n + 2
E. 5n + 7
Kudos for the right answer and explanation
The sequence formula, \(S_n = S_{n−1} + 5\), tells us that each term is 5 greater than the term before it
Let's examine some terms of the sequence....
Given: \(S_1 = 7\)
So, \(S_2 = S_1 + 5 = 7 + 5\)
\(S_3 = S_2 + 5 = (7 + 5) + 5 = 7 + 5 + 5\)
\(S_4 = 7 + 5 + 5 + 5\)
\(S_5 = 7 + 5 + 5 + 5 + 5\)
\(S_6 = 7 + 5 + 5 + 5 + 5 + 5\)
\(S_7 = 7 + 5 + 5 + 5 + 5 + 5 + 5\)
Notice that the 5th term has FOUR 5's in its sum.
The 6th term has FIVE 5's in its sum.
The 7th term has SIX 5's in its sum.
We can now generalize to say that: \(S_n = 7 + (5 + 5 + 5 + ....+5 + 5)\)
Given the pattern above, we know that there must be (n - 1) 5's in the sum.
So we can write: \(S_n = 7 + (n - 1)(5)\)
Check the answer choices..... this option does not appear.
So looks like we need to rewrite the expression.
When we expand, we get: \(S_n = 7 + 5n - 5\)
Simplify to get: \(S_n = 5n + 2\)
Answer: D
Cheers,
Brent
