1. **Stating the problem:** We have a sequence starting at 10, then subtract 4 to get 6, then multiply by 2 to get 12, and this pattern repeats. We want to find a formula for the 50th term. 2. **Understanding the pattern:** The sequence alternates between subtracting 4 and multiplying by 2. 3. **Listing the first few terms:** - $a_1 = 10$ - $a_2 = 10 - 4 = 6$ - $a_3 = 6 \times 2 = 12$ - $a_4 = 12 - 4 = 8$ - $a_5 = 8 \times 2 = 16$ - $a_6 = 16 - 4 = 12$ - $a_7 = 12 \times 2 = 24$ 4. **Observing the pattern:** Odd terms ($a_1, a_3, a_5, a_7, ...$) are obtained by multiplying the previous odd term by 2. 5. **Defining subsequences:** Let $b_n$ be the odd terms: $b_1 = 10$, $b_2 = 12$, $b_3 = 16$, $b_4 = 24$, ... 6. **Finding formula for odd terms:** From the sequence, $b_1=10$, $b_2=12$, $b_3=16$, $b_4=24$. 7. **Check ratio between odd terms:** $\frac{b_2}{b_1} = \frac{12}{10} = 1.2$, $\frac{b_3}{b_2} = \frac{16}{12} = 1.333...$, $\frac{b_4}{b_3} = \frac{24}{16} = 1.5$; not geometric. 8. **Try to express odd terms in terms of previous odd term:** Actually, odd terms are obtained by multiplying the previous odd term by 2 and then subtracting 4 once every two steps, so direct formula is complex. 9. **Alternative approach:** Since the pattern repeats every two steps, define two sequences: - Even terms: $a_{2n} = a_{2n-1} - 4$ - Odd terms: $a_{2n+1} = 2 \times a_{2n}$ 10. **Express even terms in terms of odd terms:** $a_{2n} = a_{2n-1} - 4$ 11. **Express odd terms recursively:** $a_{2n+1} = 2(a_{2n-1} - 4) = 2a_{2n-1} - 8$ 12. **Define $b_n = a_{2n-1}$ (odd terms):** Then $b_{n+1} = 2b_n - 8$ with $b_1 = 10$ 13. **Solve the recurrence:** $b_{n+1} - 2b_n = -8$ 14. **Homogeneous solution:** $b_n^h = C \times 2^{n-1}$ 15. **Particular solution:** Try constant $b_n^p = k$, substitute: $$k - 2k = -8 \Rightarrow -k = -8 \Rightarrow k = 8$$ 16. **General solution:** $$b_n = C \times 2^{n-1} + 8$$ 17. **Use initial condition $b_1=10$:** $$10 = C \times 2^{0} + 8 \Rightarrow C = 2$$ 18. **Final formula for odd terms:** $$b_n = 2 \times 2^{n-1} + 8 = 2^n + 8$$ 19. **Formula for even terms:** $$a_{2n} = a_{2n-1} - 4 = b_n - 4 = 2^n + 8 - 4 = 2^n + 4$$ 20. **Find the 50th term:** Since 50 is even, use even term formula with $n=25$: $$a_{50} = 2^{25} + 4$$ **Answer:** $$\boxed{a_{50} = 2^{25} + 4}$$