CSS Pure Mathematics Set-9

In an Arithmetic Progression, the nth term is found by starting with the first term (a1) and adding the common difference (d) exactly (n-1) times. This is expressed mathematically as a_n = a1 + (n-1)d, which is the standard definition.

The sequence consists of the cubes of consecutive natural numbers: 1^3=1, 2^3=8, 3^3=27, 4^3=64. The next term in the sequence is 5^3, which equals 125.

To find the first three terms, we substitute n = 1, 2, and 3 into the formula 2n - 4. For n=1, 2(1)-4 = -2. For n=2, 2(2)-4 = 0. For n=3, 2(3)-4 = 2. Thus, the first three terms of the sequence are -2, 0, and 2.

By mathematical definition, the arithmetic mean of two numbers 'a' and 'b' is the value 'A' such that the difference between consecutive terms is constant. Specifically, A - a = b - A, which implies that the sequence a, A, b satisfies the definition of an arithmetic sequence where the common difference is (b-a)/2.

To find the first four terms, substitute n = 1, 2, 3, and 4 into the formula. For n=1, (-1)^1(2(1)-3) = -1(-1) = 1. For n=2, (-1)^2(2(2)-3) = 1(1) = 1. For n=3, (-1)^3(2(3)-3) = -1(3) = -3. For n=4, (-1)^4(2(4)-3) = 1(5) = 5. Thus, the sequence is 1, 1, -3, 5.

To find the equivalent discount, we calculate the remaining value after both discounts. Applying a 20% discount leaves 80% (0.80), and a subsequent 15% discount on that remainder leaves 85% (0.85). Multiplying 0.80 by 0.85 equals 0.68. The total discount is 1 minus 0.68, which equals 0.32, or 32%.

To find the selling price, first calculate the discount amount: 10% of 400 is 0.10 * 400 = Rs. 40. Subtracting this discount from the original list price gives 400 - 40 = Rs. 360. Thus, the final price after the discount is applied is Rs. 360.

The sale price represents 75% of the original price (100% - 25% = 75%). To find the original price, divide the sale price of Rs. 846 by 0.75. Performing this division, 846 / 0.75 equals 1,128, which is the original price.

To determine the selling price, calculate the discount amount by multiplying the list price by the discount rate (20 * 0.05 = 1). Subtract this discount from the original list price (20 - 1 = 19). Thus, the final selling price of the article is Rs. 19.00.

To find the final price after successive discounts, apply the first discount of 20% to the original price (10 * 0.80 = 8) and then apply the second discount of 10% to the resulting value (8 * 0.90 = 7.20). Thus, the final selling price is Rs. 7.20.