What is an arithmetic progression (AP)?

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. For example, the sequence 2, 5, 8, 11 is an AP with a common difference of 3.

How do you find the nth term of an arithmetic progression?

The nth term of an arithmetic progression can be found using the formula: a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number.

What is a geometric progression (GP)?

A geometric progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 3, 6, 12, 24 is a GP with a common ratio of 2.

What is the formula for the sum of the first n terms of an arithmetic progression?

The sum of the first n terms of an arithmetic progression can be calculated using the formula: S_n = n/2 * (2a_1 + (n - 1)d), where S_n is the sum, a_1 is the first term, and d is the common difference.

If the first term of an arithmetic series is 4 and the common difference is 3, what is the 10th term?

Using the formula for the nth term: a_n = a_1 + (n - 1)d. Here, a_1 = 4, d = 3, and n = 10. So, a_10 = 4 + (10 - 1) * 3 = 4 + 27 = 31.

What is the common difference in the sequence 10, 7, 4, 1?

The common difference is found by subtracting consecutive terms. Here, d = 7 - 10 = -3, which is consistent for all pairs.

What is the formula for the nth term of a geometric progression?

The nth term of a geometric progression can be calculated using the formula: a_n = a_1 * r^(n - 1), where a_1 is the first term, r is the common ratio, and n is the term number.

How do you calculate the sum of the first n terms of a geometric progression?

The sum of the first n terms of a geometric progression is given by S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1, where a_1 is the first term and r is the common ratio.

If a geometric progression has a first term of 2 and a common ratio of 3, what is the 4th term?

Using the formula: a_n = a_1 * r^(n - 1). Here, a_1 = 2, r = 3, and n = 4. So, a_4 = 2 * 3^(4 - 1) = 2 * 27 = 54.

What is the 15th term of the arithmetic progression 5, 9, 13, ...?

The first term a_1 = 5 and the common difference d = 4. Using the formula: a_n = a_1 + (n - 1)d, a_15 = 5 + (15 - 1) * 4 = 5 + 56 = 61.

Find the sum of the first 20 natural numbers using the AP formula.

The first 20 natural numbers form an AP with a_1 = 1, d = 1, and n = 20. The sum S_20 = 20/2 * (2*1 + (20 - 1)*1) = 10 * 21 = 210.

If the sum of the first n terms of an AP is 240 and the first term is 10, with a common difference of 5, what is n?

Using the sum formula: S_n = n/2 * (2a_1 + (n - 1)d), we substitute 240 = n/2 * (2*10 + (n - 1)*5) and solve for n.

Determine the 12th term of the AP: 8, 12, 16, ...

The first term a_1 = 8 and common difference d = 4. Therefore, a_12 = 8 + (12 - 1) * 4 = 8 + 44 = 52.

What is the common ratio of the geometric progression 2, 6, 18, ...?

The common ratio r can be found by dividing any term by its preceding term. Here, r = 6/2 = 3.

What is the formula for the sum of all terms in a finite arithmetic progression when the last term is known?

The sum S can be calculated using the formula: S = n/2 * (a_1 + l), where l is the last term.