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# Algebra

-1
1
1
+171

Let a_1, a_2, a_3, \dots be a sequence. If
a_n = a_{n - 1} + a_{n - 2}
for all n \ge 3, and a_{11} = 4 and a_{10} = 1, then find a_6.

Jun 5, 2024

#1
+759
+1

We can probably write some equations with the information we have.

We have:

\(a_n = a_{n - 1} + a_{n - 2}\)  Subtracting both sides by a_n-1, we have

\(a_{n-2}=a_n-a_{n-1}\)

Now, let's plug in numbers we are given in the question. Plugging in 11 gets us \(a_9=4-1=3\)

Plugging in 10 gets us \(a_8=1-3=-2\)

Ok, let's see what happens when we plug in 9.

We get \(a_7 = a_9 - a_8\). We already know these two values!

We have \(a_7=3-(-2)=5\)

Now, we plug in 8. We get \(a_6=-2-5=-7\)

So -7 is our final answer.

Thanks! :)

Jun 5, 2024

#1
+759
+1

We can probably write some equations with the information we have.

We have:

\(a_n = a_{n - 1} + a_{n - 2}\)  Subtracting both sides by a_n-1, we have

\(a_{n-2}=a_n-a_{n-1}\)

Now, let's plug in numbers we are given in the question. Plugging in 11 gets us \(a_9=4-1=3\)

Plugging in 10 gets us \(a_8=1-3=-2\)

Ok, let's see what happens when we plug in 9.

We get \(a_7 = a_9 - a_8\). We already know these two values!

We have \(a_7=3-(-2)=5\)

Now, we plug in 8. We get \(a_6=-2-5=-7\)

So -7 is our final answer.

Thanks! :)

NotThatSmart Jun 5, 2024