Sequences & Series Assessment Test

2.

The third term of an A.P is 4x-2y and the 9th term is 10x-8y. Find the common difference.

19x-17y
x-y
8x-4y
2x

Correct Answer

A. x-y

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.

Rate this question:

3.

Evaluate (-1/2 - 1/4 + 1/8 - 1/16 +... ) -1.

2/3
0
-2/3
-1

Correct Answer

A. -2/3

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.

Rate this question:

4.

The sixth term of an arithmetical progression is half of its twelfth term. The first term is equal to...

The common difference
Half of the common difference
Zero
Double of the common difference

Correct Answer

A. The common difference

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.

Rate this question:

5.

If the nth term of an A.P is five times the 5th term, find the relationship between a and d.

A+2d=0
2a+d=0
3a+5d=0
A+3d=0

Correct Answer

A. A+3d=0

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.

Rate this question:

6.

If the harmonic mean and geometric mean of two numbers a and b are 4 and 3 2 respectively then the interval [a, b] = ___.

[3, 6]
[2, 7]
[4, 5]
[1, 8]

Correct Answer

A. [3, 6]

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].

Rate this question:

7.

If for the triangle whose perimeter is 37 cm and length of sides are in G.P. also the length of the smallest side is 9 cm, then length of remaining two sides are _ and .

12; 16
14; 14
10; 18
15; 13

Correct Answer

A. 12; 16

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.

Rate this question:

8.

If 2, b, c, 23 are in a GP, then (b - c)2 + (c - 2)2 + (23 - b)2 = ____.

625
525
441
428

Correct Answer

A. 441

9.

The coefficient of x8 in the product (x+1) (x+2) (x+3) ... (x+10) is...

1024
1300
1320
1360

Correct Answer

A. 1320

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.

Rate this question:

10.

The series 1. 1! + 2. 2! + 3. 3! + ... + n. n! = ?

(n + 1)! -n
(n + 1)! -1
N! -1 + n
N! +1 -n

Correct Answer

A. (n + 1)! -1

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.

Rate this question:

Sequences & Series Assessment Test

The first term of a G.P is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is...

Quiz Preview

The third term of an A.P is 4x-2y and the 9th term is 10x-8y. Find the common difference.

Evaluate (-1/2 - 1/4 + 1/8 - 1/16 +... ) -1.

The sixth term of an arithmetical progression is half of its twelfth term. The first term is equal to...

If the nth term of an A.P is five times the 5th term, find the relationship between a and d.

If the harmonic mean and geometric mean of two numbers a and b are 4 and 3 2 respectively then the interval [a, b] = ___.

If for the triangle whose perimeter is 37 cm and length of sides are in G.P. also the length of the smallest side is 9 cm, then length of remaining two sides are _ and .

If 2, b, c, 23 are in a GP, then (b - c)2 + (c - 2)2 + (23 - b)2 = ____.

The coefficient of x8 in the product (x+1) (x+2) (x+3) ... (x+10) is...

The series 1. 1! + 2. 2! + 3. 3! + ... + n. n! = ?

Sequences & Series Assessment Test

The first term of a G.P is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is...

Quiz Preview

The third term of an A.P is 4x-2y and the 9th term is 10x-8y. Find the common difference.

Evaluate (-1/2 - 1/4 + 1/8 - 1/16 +... ) -1.

The sixth term of an arithmetical progression is half of its twelfth term. The first term is equal to...

If the nth term of an A.P is five times the 5th term, find the relationship between a and d.

If the harmonic mean and geometric mean of two numbers a and b are 4 and 3 2 respectively then the interval [a, b] = ___.

If for the triangle whose perimeter is 37 cm and length of sides are in G.P. also the length of the smallest side is 9 cm, then length of remaining two sides are ___ and __.

If 2, b, c, 23 are in a GP, then (b - c)2 + (c - 2)2 + (23 - b)2 = ____.

The coefficient of x8 in the product (x+1) (x+2) (x+3) ... (x+10) is...

The series 1. 1! + 2. 2! + 3. 3! + ... + n. n! = ?

If for the triangle whose perimeter is 37 cm and length of sides are in G.P. also the length of the smallest side is 9 cm, then length of remaining two sides are _ and .