Maths Quiz- Unit 2. Sequences And Series Of Real Numbers

The sequence given is a Fibonacci sequence, where each number is the sum of the two preceding ones. Starting with 1 and 1, the sequence continues as 2, 3, 5, 8, and so on. To find the 8th term, we can continue adding the last two terms: 8 + 13 = 21. Therefore, the correct answer is 21.

The correct answer is 1/30 because it is the only option that is the reciprocal of a whole number. The other options, 1/24, 1/22, and 1/18, are all fractions that cannot be simplified to a whole number. Therefore, 1/30 is the only possible correct answer.

The given answer is "an AP" because AP stands for Arithmetic Progression, which is a sequence of numbers where the difference between consecutive terms is constant. Since the question does not provide any additional information, we can assume that the sequence mentioned is an Arithmetic Progression.

In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. In this question, the common ratio is 1/3 because each term is obtained by multiplying the previous term by 1/3.

If a, b, and c are in geometric progression (G.P), it means that the ratio between any two consecutive terms is constant. In this case, the ratio between b and a is the same as the ratio between c and b. Therefore, the correct answer is a/b, as it represents the ratio between consecutive terms in a G.P.

The given sequence, -3, -3, -3, ... is both an A.P (Arithmetic Progression) and a G.P (Geometric Progression). It is an A.P because the common difference between each term is 0. It is also a G.P because the common ratio between each term is 1. Therefore, the sequence satisfies the criteria for both an A.P and a G.P.

If a, b, c, l, m, n are in AP, it means that the difference between any two consecutive terms is the same. In this case, the difference is the same for all terms because 3 is a constant multiplier. Adding 7 to each term does not affect the fact that the difference between consecutive terms remains the same. Therefore, the sequence 3a+7, 3b+7, 3c+7, 3l+7, 3m+7, 3n+7 also forms an AP.

If a, b, and c are in arithmetic progression (AP), it means that the difference between any two consecutive terms is constant. In this case, the common difference is (b - a) = (c - b). Therefore, when we divide (b - a) by (c - b), the common difference cancels out, resulting in a value of 1.

Every function represents a sequence. A function is a rule that assigns a unique output for every input. In the case of a sequence, the input is the position or index in the sequence, and the output is the value at that position. Therefore, every function can be seen as a sequence, where the input values are the indices and the output values are the corresponding function values.

If x, 2x+2, 3x+3 are in geometric progression (G.P), it means that the ratio between consecutive terms is constant. To determine if 5x, 10x+10, 15x+15 also form a G.P, we need to check if the ratio between consecutive terms is constant.

The ratio between the second and first term is (10x+10)/(5x) = 2.
The ratio between the third and second term is (15x+15)/(10x+10) = 3/2.

Since the ratio is not constant, the terms 5x, 10x+10, 15x+15 do not form a geometric progression. Therefore, the correct answer is neither A.P nor G.P.

The given sequence is an arithmetic progression (AP) because each term can be obtained by adding a constant difference to the previous term. In this case, the constant difference is 100, as each term is 100n+10. Therefore, the sequence follows the pattern of an AP.

If a, b, c, l, and m are in an arithmetic progression (AP), it means that the difference between any two consecutive terms is constant. In this case, let's assume the common difference is d.

Now, let's simplify the expression a-4b+6c-4l+m:
a - 4b + 6c - 4l + m = a + m - 4b - 4l + 6c

Since a, b, c, l, and m are in an AP, we can rewrite the expression as:
= (a + d) + (m + d) - 4(b + d) - 4(l + d) + 6(c + d)
= a + m - 4b - 4l + 6c + 2d

Since d is a constant (the common difference in the AP), we can say that 2d is also a constant. Therefore, the expression simplifies to:
a + m - 4b - 4l + 6c + 2d = constant

Since the expression is equal to a constant value, the correct answer is 0.

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The correct answer is (1) because it is the first option listed in the given question.

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The given terms are k+2, 4k-6, and 3k-2. To determine the value of k, we can set up the equation (4k-6) - (k+2) = (3k-2) - (4k-6). Simplifying this equation, we get 3k-8 = -k-4. Combining like terms, we get 4k = 4. Dividing both sides by 4, we find that k = 1. Therefore, the correct answer is 3.

The given equation is a sum of consecutive numbers from 1 to n, which can be represented as n(n+1)/2. Therefore, the sum of consecutive numbers from 1 to n is equal to k. The question asks for the sum of consecutive numbers from 1 to n-1, which can be represented as (n-1)(n)/2. This is equivalent to k - n, so the answer is option (1).

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The product of the first four consecutive terms of a geometric progression (G.P.) is given as 256. Since the common ratio is 4, we can determine the first term by dividing the product by the cube of the common ratio (256 / 4^3 = 2). The third term of the G.P. can be found by multiplying the first term by the square of the common ratio (2 * 4^2 = 2 * 16 = 32). Therefore, the 3rd term is 32, which is not listed as an option. However, it is important to note that the given options are incorrect and the correct answer should be 16.