answer:Let's go step-by-step using **Solution 1** and **Solution 2** to verify the total area of the shaded regions. **Solution 1:** 1. **Calculate Total Area of the Rectangle** - The rectangle consists of twelve (1 times 1) squares. - The dimensions of the rectangle are (3 text{ units} times 4 text{ units}). - Therefore, the total area of the rectangle will be: [ 3 times 4 = 12 text{ square units}. ] 2. **Identify and Calculate the Area of the Unshaded Region** - The unshaded region is a triangle. - The base of this unshaded triangle is (1 text{ unit}) and the height is (4 text{ units}). - The area of a triangle is given by: [ text{Area of triangle} = frac{1}{2} times text{base} times text{height} = frac{1}{2} times 1 times 4 = 2 text{ square units}. ] 3. **Subtract the Area of the Unshaded Region from the Total Rectangle Area** - Total shaded area = Total rectangle area - Area of unshaded triangle: [ 12 - 2 = 10 text{ square units}. ] **Verification using Solution 2:** 1. **Calculate the Area of the Left Shaded Triangle** - The left shaded triangle has a base of (2 text{ units}) and a height of (4 text{ units}). - The area of this triangle is: [ frac{1}{2} times 2 times 4 = 4 text{ square units}. ] 2. **Calculate the Area of the Right Shaded Triangle** - The right shaded triangle has a base of (3 text{ units}) and a height of (4 text{ units}). - The area of this triangle is: [ frac{1}{2} times 3 times 4 = 6 text{ square units}. ] 3. **Add the Areas of the Two Triangles** - Total shaded area = Area of left triangle + Area of right triangle: [ 4 + 6 = 10 text{ square units}. ] # Conclusion: Both solution methods confirm that the total area of the shaded regions is (10) square units. (boxed{10})

answer:The area of Square C, which has side lengths of y inches, is calculated by: [ text{Area of Square C} = y cdot y = y^2 text{ square inches.} ] The area of Square D, with side lengths of 5y inches, is: [ text{Area of Square D} = 5y cdot 5y = 25y^2 text{ square inches.} ] The ratio of the area of Square C to the area of Square D is: [ text{Ratio} = frac{y^2}{25y^2} = frac{1}{25} ] Thus, the ratio of the area of Square C to Square D is boxed{frac{1}{25}}.

answer:The banker's gain is the difference between the amount due at the end of the period and the present worth of that amount. The banker's gain can also be calculated using the formula: Banker's Gain (BG) = (Amount Due (AD) - Present Worth (PW)) = (Simple Interest (SI) on PW) Given that the banker's gain is Rs. 24 and the rate of interest is 10% per annum, we can use the formula for simple interest to find the present worth. The formula for simple interest is: SI = (P × R × T) / 100 Where: SI = Simple Interest P = Principal (Present Worth) R = Rate of Interest per annum T = Time in years We know that the banker's gain (BG) is equal to the simple interest (SI) on the present worth (PW), and the time (T) is 2 years. So we can write: BG = (PW × R × T) / 100 Now we can plug in the values we know: 24 = (PW × 10 × 2) / 100 To find the present worth (PW), we can solve for PW: 24 = (PW × 20) / 100 24 = PW × 0.2 PW = 24 / 0.2 PW = 120 Therefore, the present worth of the sum is Rs. boxed{120} .

answer:1. **Expression for ( g(n+2) )**: [ g(n+2) = dfrac{3+2sqrt{3}}{6}left(dfrac{1+sqrt{3}}{2}right)^{n+2} + dfrac{3-2sqrt{3}}{6}left(dfrac{1-sqrt{3}}{2}right)^{n+2} ] Using the identity ( left(dfrac{1+sqrt{3}}{2}right)^{n+2} = left(dfrac{1+sqrt{3}}{2}right)^n left(dfrac{1+sqrt{3}}{2}right)^2 ) and similarly for ( left(dfrac{1-sqrt{3}}{2}right)^{n+2} ): [ g(n+2) = dfrac{3+2sqrt{3}}{6} cdot left(dfrac{3+2sqrt{3}}{4}right) cdot left(dfrac{1+sqrt{3}}{2}right)^n + dfrac{3-2sqrt{3}}{6} cdot left(dfrac{3-2sqrt{3}}{4}right) cdot left(dfrac{1-sqrt{3}}{2}right)^n ] Simplifying coefficients: [ g(n+2) = dfrac{3}{4} left(dfrac{1+sqrt{3}}{2}right)^n + dfrac{3}{4} left(dfrac{1-sqrt{3}}{2}right)^n = dfrac{3}{4} g(n) ] 2. **Computing ( g(n+2) - g(n) )**: [ g(n+2) - g(n) = dfrac{3}{4} g(n) - g(n) = dfrac{-1}{4} g(n) ] 3. **Conclusion**: [ -dfrac{1{4} g(n)} ] boxed{The final answer is (boxed{-dfrac{1}{4} g(n)})}