The sequence S is defined by \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\) for each integer...

[BTGmoderatorLU]

Source: Manhattan Prep

The sequence S is defined by \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\) for each integer \(n \geq 4\). If \(S_1=4, S_2=0,\) and \(S_4=-4\), what is the value of \(S_6?\)

A. -2
B. -12
C. -16
D. -20
E. -24

The OA is E

[Jay@ManhattanReview]

Source: Manhattan Prep

The sequence S is defined by \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\) for each integer \(n \geq 4\). If \(S_1=4, S_2=0,\) and \(S_4=-4\), what is the value of \(S_6?\)

A. -2
B. -12
C. -16
D. -20
E. -24

The OA is E

\(S_6=S_5+S_4+S_3-5\)

\(S_6=(S_4+S_3+S_2-5)+S_4+S_3-5\)

\(S_6=2S_4+2S_3+S_2-5-5\) ---(1)

Since \(n \geq 4\), we can apply \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\) to get the value of \(S_3\).

So let's find out the value of \(S_3\).

From \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\), we have

\(S_4=S_3+S_2+S_1-5\)

Plugging-in the values, we have

\(-4=S_3+0+4-5\)

\(S_3=-3\)

Coming back to equation #1.

\(S_6=2S_4+2S_3+S_2-5-5\)

\(S_6=2*-4+2*-3+0-5-5\)

\(S_6=-24\)

The correct answer: E

Hope this helps!

-Jay
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[swerve]

Source: Manhattan Prep

The sequence S is defined by \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\) for each integer \(n \geq 4\). If \(S_1=4, S_2=0,\) and \(S_4=-4\), what is the value of \(S_6?\)

A. -2
B. -12
C. -16
D. -20
E. -24

The OA is E

Let's find \(S_3\)

\(S_4=S_3+S_2+S_1-5\). Substitute values in the equation.

Then \(S_3=-3\)

\(S_5=S_4+S_3+S_2-5\). No need to get the value

\(S_6=S_5+S_4+S_3-5\). Substitute \(S_5\) from above then

\(S_6=2S_4+S_3+S_2-10 = -8 - 6 +0 -10 =-24\)

Answer: E

[Scott@TargetTestPrep]

Source: Manhattan Prep

The sequence S is defined by \(S_n=S_{n-1}+S_{n-2}+S_{n-3}-5\) for each integer \(n \geq 4\). If \(S_1=4, S_2=0,\) and \(S_4=-4\), what is the value of \(S_6?\)

A. -2
B. -12
C. -16
D. -20
E. -24

The OA is E

To find the value of s(6), we need the values of s(5), s(4) and s(3). Since we are already given the value of s(4), we need to determine the values of s(5) and s(3).

For s(3), we have:

s(4) = s(3) + s(2) + s(1) - 5

-4 = s(3) + 0 + 4 - 5

-3 = s(3)

So now for s(5), we have:

s(5) = s(4) + s(3) + s(2) - 5

s(5) = -4 + (-3) + 0 - 5

s(5) = -12


Finally, for s(6), we have:

s(6) = s(5) + s(4) + s(3) - 5

s(6) = -12 + (-4) + (-3) - 5 = - 24

Answer: E