answer:Compute the first few terms to identify a pattern: [ begin{align*} y_2 &= frac{(m+1) cdot m - (m - 0) cdot 1}{2} = frac{m^2 + m - m}{2} = frac{m^2}{2}, y_3 &= frac{(m+1) frac{m^2}{2} - (m - 1) cdot m}{3} = frac{(m+1)m^2 - 2(m - 1)m}{6} = frac{m(m+1)(m-1)}{6}. end{align*} ] Assume the pattern: [y_k = frac{(m+1)(m)(m-1) dotsm (m-k+2)}{(k)!}] for k ge 2. Use mathematical induction to prove this. Base cases (k=2, 3) as computed above hold. Assume true for k = i and k = i+1. Then: [ begin{align*} y_{i+2} &= frac{(m+1) y_{i+1} - (m - i) y_i}{i + 2} &= frac{(m+1)frac{m(m-1)dotsm(m-i+1)}{(i+1)!} - (m - i)frac{m(m-1)dotsm(m-i+2)}{i!}}{i + 2} &= frac{m(m-1)dotsm(m-i+2)}{i!} cdot frac{(m+1)(m-i+1) - (m - i)(i+1)}{i+2} &= frac{m(m-1)dotsm(m-i+2)(m-i+1)(m-i)}{(i+2)!}. end{align*} ] This completes the induction. For k le m+1, y_k = binom{m+1}{k}, and y_k = 0 for k > m+1. Summing these yields: [ y_0 + y_1 + y_2 + dotsb = binom{m+1}{0} + binom{m+1}{1} + dots + binom{m+1}{m+1} = 2^{m+1}. ] The final answer is boxed{2^{m+1}}.

answer:To find the sum of the intercepts on the x-axis and y-axis, we set the other variable to zero and solve for the remaining variable. For the x-intercept, set y = 0: 2x - 3(0) - 6k = 0 implies x = 3k. For the y-intercept, set x = 0: 2(0) - 3y - 6k = 0 implies y = -2k. The sum of the intercepts is the sum of x when y=0 and y when x=0: 3k + (-2k) = k. According to the problem, this sum should be equal to 1: k = 1. Therefore, the value of k is (boxed{1}).

answer:1. **Understanding the conversion**: 1 euro equals 1.2 dollars. 2. **Including the discount**: Purchasing with euros gives a 10% discount. Thus, M euros, translating to 1.2M dollars, effectively amounts to 1.2M times 1.1 dollars considering the discount factor (since paying 10% less means getting more value by a factor of 1.1). 3. **Setting up the proportion**: Given P bottles for R dollars, the number of bottles y that can be purchased with 1.2M times 1.1 dollars is derived by: [ frac{P text{ bottles}}{R text{ dollars}} = frac{y text{ bottles}}{1.2M times 1.1 text{ dollars}} ] Solving for y we have: [ y cdot R = P cdot 1.2M times 1.1 ] [ y = frac{P cdot 1.2M times 1.1}{R} ] 4. **Simplify the expression**: Simplifying, we get: [ y = frac{1.32PM}{R} ] 5. **Check units and validity**: The units match, as the expression simplifies to the number of bottles (dimensionless dollars cancel out). Conclusion: The number of bottles of water that can be purchased for M euros is frac{1.32PM{R}}. The final answer is boxed{A}.

answer:**Answer**: The anther culture of diploid corn results in haploid plants, not diploids, so ① is correct, and ⑧ is incorrect; For tetraploids, a haploid contains 2 sets of chromosomes, having 10 pairs of homologous chromosomes, so ② is incorrect, and ④ is correct; A haploid with 2 sets of chromosomes can undergo meiosis to form fertile gametes, so ③ is incorrect, and ⑤ is correct; The genotype of diploid corn is Aa. After treatment with colchicine, the genotype becomes AAaa. The produced pollen can be AA, aa, Aa. The haploids obtained from anther culture can be AA, aa, Aa, having both homozygous and heterozygous, so ⑥ is correct, and ⑦ is incorrect. Therefore, the correct answer is boxed{A}. **Analysis**: Assuming the genotype of the diploid corn is Aa, after treatment with colchicine, the genotype becomes AAaa. The produced pollen can be AA, aa, Aa. The haploids obtained from anther culture can be AA, aa, Aa, having both homozygous and heterozygous. These haploid plants have homologous chromosomes and can produce fertile gametes.