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.
52
Identify the next term in the sequence: 1, 8, 27, 64, ...
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.
53
Given the sequence defined by a_n = 2n - 4, what are the first three terms of the sequence?
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.
54
For any two numbers 'a' and 'b', a value 'A' is defined as the arithmetic mean between them if the sequence a, A, b forms which of the following?
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.
55
Determine the first four terms of the sequence defined by the formula a_n = (-1)^n(2n-3).
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.
56
Calculate the single equivalent discount for two successive discounts of 20% and 15%.
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%.
57
A book has a list price of Rs. 400. If a 10% discount is applied, what is the final selling price?
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.
58
A suitcase is sold for Rs. 846 after a 25% discount. What was the original retail price of the suitcase?
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.
59
An item has a list price of Rs. 20.00. If a 5% discount is applied, what is the final selling 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.
60
Calculate the final selling price of an item with a list price of Rs. 10.00 after applying successive discounts of 20% and 10%.
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.
