Question

Can you please help, I'm offering 30 points!
if a1=5 and an=nan-1+2 then find the value of a4

Answer 1

a₄ = 154

Explanation:

You want the value of a₄ given the recursive sequence definition ...

`\begin{cases}a_1=5\\a_(n)=n\cdot a_(n-1)+2\end{cases}`

Solution

The easiest way to find the first few terms of a recursively-defined sequence is to use the recursive definition directly:

`a_2=2\cdot a_1 +2 = 2(5) +2 = 12\\\\a_3=3\cdot a_2+2=3(12)+2 =38\\\\a_4=4\cdot a_3+2=4(38)+2=154`

The value of a₄ is 154.

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Additional comment

The terms of this sequence get rapidly larger. The 10th term is 23,357,002. The general term can be defined in terms of the Euler gamma function and the incomplete gamma function:

`a_n=\Gamma(n+1)+2e\Gamma(n+1,1)`

Answer 2

`a_4 = 154`

Explanation:

In this problem, we are asked to find the fourth term (
`\square_4`) of a sequence
`a` given that
`a_1 = 5` and
`a_n = na_(n-1) + 2`.

To solve for this, we can first find the second term in the sequence by inputting
`a_2` into the sequence definition.

`a_2 = 2a_(2-1) + 2`

Then, we can replace
`a_(2-1)` with the value that we were given:
`a_1 = 5`.

`a_2 = 2(5) + 2`

`a_2 = 10 + 2`

`a_2 = 12`

Next, we can solve for the third term in the sequence.

`a_3 = 3a_(3-1) + 2`

And we can replace
`a_(3-1)` with the value that we just solved for:
`a_2 = 12`.

`a_3 = 3(12) + 2`

`a_3 = 36 + 2`

`a_3 = 38`

Finally, we can solve for fourth term in the sequence.

`a_4 = 4a_(4-1) + 2`

We can again replace
`a_(4-1)` with the value that we just solved for:
`a_3 = 38`.

`a_4 = 4(38) + 2`

`a_4 = 152 + 2`

`a_4 = 154`