In a sequence of 10 terms, the first term is 1 , the second term is $x$, and each term after the second is the sum of the previous two terms. For example, if $x=11$, the sequence would be $1,11,12,23,35,58,93,151,244,395$. For some values of $x$, the number 463 appears in the sequence. If $x$ is a positive integer, what is the sum of all the values of $x$ for which 463 appears in the sequence?
(A) 1156
(B) 1296
(C) 1248
(D) 1528
(E) 1283
We are given that the first two terms of a 10 term sequence are 1 and $x$.

Since each term after the second is the sum of the previous two terms, then the third term

is $1+x$.

Since the fourth term is the sum of the second and third terms, then the fourth term

is $x+(1+x)=1+2 x$.

Continuing in this manner, we construct the 10 term sequence:

$$
1, x, 1+x, 1+2 x, 2+3 x, 3+5 x, 5+8 x, 8+13 x, 13+21 x, 21+34 x
$$

Each of the second through tenth terms is dependent on the value of $x$, and thus, any one of these terms could potentially equal 463 .

For the second term to equal 463 , we need $x=463$, which is possible since the only requirement is that $x$ is a positive integer.

Thus, if $x=463$ then 463 appears as the second term in the sequence.

For the third term to equal 463 , we need $1+x=463$, or $x=462$.

Thus, if $x=462$ then 463 appears as the third term in the sequence.

For the fourth term to equal 463 , we need $1+2 x=463$, or $2 x=462$ or $x=231$.

Thus, if $x=231$ then 463 appears as the fourth term in the sequence.

For the fifth term to equal 463 , we need $2+3 x=463$, or $3 x=461$ or $x=\frac{461}{3}$.

However, $\frac{461}{3}$ is not an integer, and thus, 463 cannot appear as the fifth term in the sequence. We continue in this manner and summarize all the results in the table below.

| Term | Expression | Equation | Value of $x$ | Is $x$ an integer? |
| :---: | :---: | :---: | :---: | :---: |
| 2nd | $x$ | $x=463$ | $x=463$ | Yes |
| 3 rd | $1+x$ | $1+x=463$ | $x=462$ | Yes |
| 4 th | $1+2 x$ | $1+2 x=463$ | $x=231$ | Yes |
| 5 th | $2+3 x$ | $2+3 x=463$ | $x=\frac{461}{3}$ | No |
| 6 th | $3+5 x$ | $3+5 x=463$ | $x=92$ | Yes |
| 7 th | $5+8 x$ | $5+8 x=463$ | $x=\frac{458}{8}$ | No |
| 8 th | $8+13 x$ | $8+13 x=463$ | $x=35$ | Yes |
| 9 th | $13+21 x$ | $13+21 x=463$ | $x=\frac{450}{21}$ | No |
| 10 th | $21+34 x$ | $21+34 x=463$ | $x=13$ | Yes |

Therefore, the sum of all possible integer values of $x$ for which 463 appears in the sequence is $463+462+231+92+35+13=1296$.