Question 1168274
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The recursive formula says we get the next term in the sequence by doubling the current term and adding 5.  Starting with 3 as the first term...<br>
a(1) = 3
a(2) = 3(2)+5 = 11
a(3) = 11(2)+5 = 27
a(4) = 27(2)+5 = 59
a(5) = 59(2)+5 = 123<br>
It would be very tedious to find the 50th term by continuing to use the recursive formula, so to find the 50th term we would like to find an explicit formula for the n-th term.<br>
To do that, we could try the method of finite differences.  Here are the first few terms of the sequence and the first and second differences:<br><pre>

    3   11    27   59  123
      8    16   32   64
         8    16   32</pre>
That pattern of differences tells us that the formula for the n-th term is not a polynomial but instead is based on powers of 2.<br>
Playing with the first few numbers in the sequence, we see that adding 5 to each term gives us a sequence of powers of 2:<br>
8, 16, 32, 64, 128, ...<br>
So...
a(1) is 8-5 = 2^3-5
a(2) is 16-5 = 2^4-5
a(3) is 32-5 = 2^5-5
...<br>
And we see the pattern for the n-th term is<br>
a(n) = 2^(n+2)-5<br>
So the 50th term is<br>
2^52-5 = 9007199254740987<br>
My TI-84 calculator won't calculate that number; and I tried using excel but it fell one digit short of being able to calculate the exact number. I used an online program (pari) to find the exact answer.<br>