Arithmetic Progression

Define an Arithmetic Progression (A.P).

• A sequence of numbers that follows a geometric pattern.

• A sequence of numbers that only includes even numbers.

• An Arithmetic Progression (A.P) is a sequence of numbers with a constant difference between consecutive terms.

• A sequence of numbers with a variable difference between terms.

Answer explanation

An Arithmetic Progression (A.P) is defined by a constant difference between consecutive terms, distinguishing it from other sequences. The correct choice accurately describes this fundamental property.

What is the common difference in an A.P.?

• The common difference is the sum of the first and last terms.

• The common difference is the average of all terms in the A.P.

• The common difference in an A.P. is the difference between consecutive terms.

• The common difference is the product of the first two terms.

Answer explanation

The common difference in an A.P. (Arithmetic Progression) is defined as the difference between any two consecutive terms. This means that if you subtract one term from the next, you will always get the same value, which is the common difference.

If the first term of an A.P. is 5 and the common difference is 3, what is the 10th term?

• 25

• 40

• 32

• 15

Answer explanation

The nth term of an A.P. is given by the formula: a_n = a + (n-1)d. Here, a = 5, d = 3, and n = 10. So, a_{10} = 5 + (10-1)3 = 5 + 27 = 32. Thus, the 10th term is 32.

Write the formula for the nth term of an A.P.

• a_n = a / (n - 1) + d

• a_n = a - (n + 1)d

• a_n = a * n + d

• a_n = a + (n - 1)d

Answer explanation

The formula for the nth term of an arithmetic progression (A.P.) is given by a_n = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference. This correctly represents the sequence of terms in an A.P.

If the first term is 2 and the last term is 20, with 10 terms in total, what is the common difference?

• 4

• 5

• 2

• 10

Answer explanation

To find the common difference (d) in an arithmetic sequence, use the formula: d = (last term - first term) / (number of terms - 1). Here, d = (20 - 2) / (10 - 1) = 18 / 9 = 2. Thus, the common difference is 2.

Find the sum of the first 15 terms of the A.P. where the first term is 4 and the common difference is 2.

• 250

• 270

• 180

• 300

Answer explanation

The sum of the first n terms of an A.P. is given by S_n = n/2 * (2a + (n-1)d). Here, a = 4, d = 2, and n = 15. Thus, S_15 = 15/2 * (2*4 + (15-1)*2) = 15/2 * (8 + 28) = 15/2 * 36 = 270.

If the 5th term of an A.P. is 20 and the 10th term is 35, find the first term.

• 10

• 5

• 8

• 15

Answer explanation

In an A.P., the nth term is given by a_n = a + (n-1)d. For the 5th term (20): a + 4d = 20. For the 10th term (35): a + 9d = 35. Solving these equations gives a = 8. Thus, the first term is 8.

How many terms are there in the A.P. 3, 7, 11, ..., 51?

• 10

• 12

• 13

• 15

Answer explanation

The A.P. has a first term of 3 and a common difference of 4. To find the number of terms, use the formula: n = (last term - first term) / common difference + 1. Here, n = (51 - 3) / 4 + 1 = 13. Thus, the answer is 13.

If the sum of the first n terms of an A.P. is 150 and the first term is 10, what is the common difference?

• 5

• 20

• 30

• 1.1111

Answer explanation

The sum of the first n terms of an A.P. is given by S_n = n/2 * (2a + (n-1)d). Here, S_n = 150, a = 10. Solving for d gives d = (150 - 10n)/n + 10. For n=15, d = 1.1111, which is the correct answer.

What is the relationship between the first term, last term, and the number of terms in an A.P. for calculating the sum?

• The sum of an A.P. is calculated by multiplying the first term by the number of terms.

• The sum of an A.P. can be calculated using only the last term.

• The sum of an A.P. is only dependent on the first term.

• The sum of an A.P. is related to the first term, last term, and number of terms by the formula S_n = n/2 * (a + l).

Answer explanation

The correct choice states that the sum of an A.P. is calculated using the formula S_n = n/2 * (a + l), which incorporates the first term (a), last term (l), and the number of terms (n). This formula is essential for finding the sum.