Sequences & Series Assessment Test

Explanation
In a geometric progression (G.P), the sum to infinity can be calculated using the formula S = a/(1 - r), where S is the sum to infinity, a is the first term, and r is the common ratio. In this question, it is given that the first term is twice the common ratio. Let's assume the first term is 2x and the common ratio is x. Therefore, the sum to infinity can be written as 2x/(1 - x). To find the sum of the first two terms, we substitute the values of the first term and common ratio into the formula, which gives us 2x + 2x^2/(1 - x). Simplifying this expression, we get (2x - 2x^2 + 2x^2)/(1 - x) = 2x/(1 - x). Since the sum to infinity is given as 8/5, we can equate it with 2x/(1 - x) and solve for x. By substituting the value of x back into the expression for the sum of the first two terms, we get the answer as 72/25.

Explanation
The common difference in an arithmetic progression (A.P) is the constant value that is added or subtracted to each term in order to obtain the next term. In this case, the third term is given as 4x-2y and the ninth term is given as 10x-8y. To find the common difference, we need to determine the difference between any two consecutive terms in the A.P. By subtracting the third term from the ninth term, we get (10x-8y) - (4x-2y) = 6x-6y. Therefore, the common difference is x-y.

Explanation
The given expression is a geometric series with a common ratio of -1/2. The sum of an infinite geometric series is given by the formula S = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = -1/2 and r = -1/2. Plugging these values into the formula, we get S = (-1/2) / (1 - (-1/2)) = (-1/2) / (3/2) = -1/3. Finally, subtracting 1 from -1/3 gives us the answer of -2/3.

Explanation
In an arithmetic progression, each term is obtained by adding a constant value, called the common difference, to the previous term. In this question, it is given that the sixth term is half of the twelfth term. This implies that the difference between the sixth term and the first term is equal to the difference between the twelfth term and the sixth term, which is the common difference. Therefore, the first term is equal to the common difference.

Explanation
The given information states that the nth term of an arithmetic progression (A.P) is five times the 5th term. Using the formula for the nth term of an A.P, we can write this as a + (n-1)d = 5(a + 4d), where a is the first term and d is the common difference. Simplifying this equation, we get a + nd - d = 5a + 20d. Rearranging the terms, we have nd - 5a = 20d - d, which can be further simplified to nd - 5a = 19d. Rearranging the terms again, we get nd = 19d + 5a. Dividing both sides by d, we have n = 19 + 5a/d. Therefore, the relationship between a and d is given by a + 3d = 0.

Explanation
The harmonic mean of two numbers is the reciprocal of the arithmetic mean of their reciprocals. The geometric mean of two numbers is the square root of their product. In this case, the harmonic mean of a and b is 4, so the arithmetic mean of their reciprocals is 1/4. Therefore, the reciprocals of a and b are 1/4 and 1/4 respectively. The geometric mean of a and b is 3√2, so their product is (3√2)^2 = 18. Since the reciprocals of a and b are 1/4 and 1/4 respectively, their product is 1/4 * 1/4 = 1/16. Therefore, a*b = 18 and a*b = 1/16. Solving these equations, we find that a = 3 and b = 6. Therefore, the interval [a, b] is [3, 6].

Explanation
The length of the sides of a triangle in a geometric progression (G.P.) can be found by multiplying the common ratio to the previous term. Since the length of the smallest side is 9 cm, we can find the other two sides by multiplying 9 by the common ratio. If the length of the remaining two sides are 12 cm and 16 cm, then the common ratio must be 12/9 = 4/3. Therefore, the length of the remaining two sides are 12 cm and 16 cm.

Explanation
To find the coefficient of x^8 in the product (x+1)(x+2)(x+3)...(x+10), we need to consider terms that contribute to x^8 when multiplied together. The term x^8 can be obtained by multiplying x^4 from (x+4) and x^4 from (x+8). The coefficient of x^8 is the product of the coefficients of these terms, which is 4 * 8 = 32. However, this is not the final answer. Since we have 10 terms in total, we need to consider all possible combinations that result in x^8. By using the binomial coefficient formula, we can calculate the coefficient to be 10C4 * 32 = 210 * 32 = 6720. Therefore, the correct answer is 1320.

Explanation
The given series is the sum of factorials of consecutive numbers from 1 to n. To find a pattern in the series, let's take a few terms as an example:
For n = 1, the series is 1! = 1
For n = 2, the series is 1! + 2! = 1 + 2 = 3
For n = 3, the series is 1! + 2! + 3! = 1 + 2 + 6 = 9
From the pattern, we can observe that each term in the series is the sum of the previous term and the factorial of the next number. So, the general formula for the series is (n + 1)! - 1.