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question:Consider the function ( g(x) = 5x 7 ). For what value of ( a ) is ( g(a) = 0 )?
answer:Given the function ( g(x) = 5x - 7 ), we want to find the value of ( a ) for which ( g(a) = 0 ). 1. Set up the equation from the function: [ g(a) = 5a - 7 ] 2. Equate this to zero and solve for ( a ): [ 5a - 7 = 0 ] 3. Solve for ( a ): [ 5a = 7 ] [ a = frac{7}{5} ] 4. Thus, the value of ( a ) is ( boxed{frac{7}{5}} ).
question:The H1N1 influenza virus is a parasite that lives inside the host's cells. If there are initially 2 cells, denoted as a=2, they divide according to the following rule: after 1 hour, they divide into 4 cells and 1 dies, after 2 hours, they divide into 6 cells and 1 dies, after 3 hours, they divide into 10 cells and 1 dies, ..., let the number of cells after n hours be a_n, then a_n= （expressed in terms of n）.
answer:Following the pattern, a_1=4-1=3, a_2=2times3-1=5, a_3=2times5-1=9, ..., a_{n+1}=2a_n-1; Therefore, a_{n+1}-1=2(a_n-1), which means {a_n-1} is a geometric sequence with the first term being 2 and the common ratio being 2, So, a_n-1=2^n, therefore a_n=2^n+1. Hence, the answer is: boxed{2^n+1}.
question:Let ( r_{1}, r_{2}, r_{3}, r_{4} ) be the four roots of the polynomial ( x^{4}4 x^{3}+8 x^{2}7 x+3 ). Find the value of [ frac{r_{1}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+frac{r_{2}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+frac{r_{3}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+frac{r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}} ]
answer:1. Let ( r_1, r_2, r_3, r_4 ) be the four roots of the polynomial ( x^4 - 4x^3 + 8x^2 - 7x + 3 ). 2. Using Vieta's formulas, we know the sums and products of the roots: - The sum of the roots is given by the coefficient of ( x^3 ), so ( r_1 + r_2 + r_3 + r_4 = 4 ). - The sum of the products of the roots taken two at a time is given by the coefficient of ( x^2 ), so [ r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4 = 8. ] 3. We are asked to find the value of: [ frac{r_1^2}{r_2^2 + r_3^2 + r_4^2} + frac{r_2^2}{r_1^2 + r_3^2 + r_4^2} + frac{r_3^2}{r_1^2 + r_2^2 + r_4^2} + frac{r_4^2}{r_1^2 + r_2^2 + r_3^2}. ] 4. Add ( 1 ) to each fraction inside the expression: [ frac{r_1^2}{r_2^2 + r_3^2 + r_4^2} + 1 + frac{r_2^2}{r_1^2 + r_3^2 + r_4^2} + 1 + frac{r_3^2}{r_1^2 + r_2^2 + r_4^2} + 1 + frac{r_4^2}{r_1^2 + r_2^2 + r_3^2} + 1 ] [ = frac{r_1^2 + r_2^2 + r_3^2 + r_4^2}{r_2^2 + r_3^2 + r_4^2} + frac{r_1^2 + r_2^2 + r_3^2 + r_4^2}{r_1^2 + r_3^2 + r_4^2} + frac{r_1^2 + r_2^2 + r_3^2 + r_4^2}{r_1^2 + r_2^2 + r_4^2} + frac{r_1^2 + r_2^2 + r_3^2 + r_4^2}{r_1^2 + r_2^2 + r_3^2} ] 5. We need to calculate ( r_1^2 + r_2^2 + r_3^2 + r_4^2 ): - Recall that: [ (r_1 + r_2 + r_3 + r_4)^2 = r_1^2 + r_2^2 + r_3^2 + r_4^2 + 2(r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4) ] - Substituting the known sums: [ 4^2 = r_1^2 + r_2^2 + r_3^2 + r_4^2 + 2 times 8 ] [ 16 = r_1^2 + r_2^2 + r_3^2 + r_4^2 + 16 ] [ r_1^2 + r_2^2 + r_3^2 + r_4^2 = 0 ] 6. Substituting ( r_1^2 + r_2^2 + r_3^2 + r_4^2 = 0 ) into the expression: [ frac{0}{r_2^2 + r_3^2 + r_4^2} + frac{0}{r_1^2 + r_3^2 + r_4^2} + frac{0}{r_1^2 + r_2^2 + r_4^2} + frac{0}{r_1^2 + r_2^2 + r_3^2} = 0 ] 7. Recalling that we added ( 1 ) to each term, we subtract the number of terms we added: [ 0 - 4 = -4 ] Conclusion: ( boxed{-4} )
question:m fetched a total simple interest of 4016.25 at the rate of 9 % . p . a . in 5 years . what is the sum ?
answer:To find the sum, we can use the simple interest formula: Simple Interest (SI) = Principal (P) * Rate (R) * Time (T) / 100 We are given: Simple Interest (SI) = 4016.25 Rate (R) = 9% per annum Time (T) = 5 years We need to find the Principal (P). Rearranging the formula to solve for P, we get: P = SI / (R * T / 100) Now, we can plug in the given values: P = 4016.25 / (9 * 5 / 100) P = 4016.25 / (45 / 100) P = 4016.25 / 0.45 P = 8925 Therefore, the sum (Principal) is boxed{8925} .