The Arithmetic Progression Test! Math Quiz

Explanation
The common difference of an arithmetic progression (A.P.) is the constant value by which each term increases or decreases. In this case, the general term of the A.P. is given by an = 2n + 1. By comparing the terms, we can see that each term increases by 2. Therefore, the common difference of this A.P. is 2.

Explanation
The given arithmetic progression (A.P.) starts with 2 and has a common difference of 3. To find the number of terms, we can use the formula for the nth term of an A.P., which is given by: nth term = a + (n-1)d, where a is the first term and d is the common difference. In this case, a = 2 and d = 3. We need to find the value of n when the nth term is 59. Plugging in these values into the formula, we get: 59 = 2 + (n-1)3. Simplifying this equation, we get: 57 = (n-1)3. Dividing both sides by 3, we get: 19 = n-1. Adding 1 to both sides, we get: 20 = n. Therefore, the number of terms in the A.P. is 20.

Explanation
The given arithmetic progression (A.P.) starts with -11 and has a common difference of 3. To find the first positive term, we need to find the first term in the A.P. that is greater than 0. Starting from -11, the next term is -8, then -5, and so on. The first positive term is 1, which is greater than 0. Therefore, the answer is 1.

Explanation
The sum of the first 20 odd natural numbers can be calculated by using the formula n^2, where n is the number of terms. In this case, n is 20. So, the sum would be 20^2 = 400.

Explanation
The sum of the first 20 natural numbers can be calculated using the formula for the sum of an arithmetic series. The formula is given by (n/2)(a + l), where n is the number of terms, a is the first term, and l is the last term. In this case, the first term is 1 and the last term is 20. Plugging these values into the formula gives (20/2)(1 + 20) = 10(21) = 210. Therefore, the correct answer is 210.

Explanation
The sum of the first 10 multiples of 7 can be calculated by multiplying 7 by each of the numbers from 1 to 10 and then adding them together. In this case, the sum would be 7 + 14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 + 70 = 385.

Explanation
The given arithmetic progression (A.P.) starts with 3 and has a common difference of 4. To find the number of terms, we can use the formula for the sum of an A.P., which is Sn = (n/2)(2a + (n-1)d), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference. Plugging in the values, we have 210 = (n/2)(2(3) + (n-1)(4)). Simplifying this equation gives 210 = (n/2)(6 + 4n - 4). Further simplification gives 210 = (n/2)(4n + 2). Solving for n, we get n = 10. Therefore, the number of terms is 10.

Explanation
The sum of the first and the tenth term of an arithmetic progression (A.P.) is given as 50. In an A.P., the sum of the terms can be found by using the formula: n/2 * (2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. Here, we have n = 10 (as we need to find the sum of the first 10 terms), and a1 + a10 = 50. Plugging these values into the formula, we get: 10/2 * (2a + 9d) = 50. Simplifying this equation gives us 5(2a + 9d) = 50, which further simplifies to 2a + 9d = 10. Since we need to find the sum of the first 10 terms, we can use the formula: S10 = 10/2 * (2a + (10-1)d) = 5(2a + 9d) = 5 * 10 = 50. Therefore, the value of a1 + a2 + ... + a10 is 250.

Explanation
In an arithmetic progression (A.P.), the difference between consecutive terms is constant. In this case, the common difference is -4, as each term is obtained by subtracting 4 from the previous term. To find the first negative term, we need to find the term where the value becomes negative. Starting from the first term (121) and subtracting 4 successively, we find that the term 32 is the first negative term in the A.P.

Explanation
The given sequence is an arithmetic progression with a common difference of 8. To find the number of terms required to give a sum of 636, we can use the formula for the sum of an arithmetic progression: Sn = (n/2)(2a + (n-1)d), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference. Plugging in the values, we have 636 = (n/2)(2*9 + (n-1)*8). Simplifying the equation, we get 636 = (n/2)(26 + 8n - 8). Solving for n, we find that n = 12. Therefore, the number of terms required to give a sum of 636 is 12.

Explanation
The given arithmetic progression (A.P.) is: 2, 5, 8, ..., 35
The common difference (d) between consecutive terms is 5 - 2 = 3.
Now, we want to find the 4th term from the end. To do this, we'll work backward from the last term (35).
The last term (n) is 35. We want to find the 4th term from the end, so we'll subtract 3 times the common difference from 35, because there are 4 terms between the last term and the term we're looking for.
n = 35 - 3 * 3 n = 35 - 9 n = 26
So, the 4th term from the end of the A.P. is indeed 26.

Explanation
The common difference of an arithmetic progression (A.P.) is the constant value that is added to each term to get the next term. In this case, the 11th term is 35 and the 13th term is 41. To find the common difference, we subtract the 11th term from the 13th term. 41 - 35 = 6. Therefore, the common difference of the A.P. is 6.

Explanation
To find the next term in the arithmetic progression (A.P.) represented by the sequence of square roots, you need to identify the common difference between consecutive terms.
Let's first calculate the common difference (d) for this sequence:
The first term: √8
The second term: √18
To find the common difference, subtract the first term from the second term:
Common Difference (d) = √18 - √8
Now, simplify:
d = √(9 * 2) - √(4 * 2)
d = (3√2) - (2√2)
d = √2
Now that we have the common difference (d = √2), we can find the next term in the sequence:
The third term: √32
To find the next term:
The fourth term = (third term) + (common difference) = √32 + √2 = 4√2 + √2 = 5√2
So, the next term in the sequence is 5√2.

Explanation
To find the multiples of 4 between 10 and 250, you can start by dividing 10 by 4 to find the first multiple, and then continue with increments of 4 until you reach 250. Here are the multiples of 4 in that range:
12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248