Recursive Formula - Rule of Arithmetic and Geometric Sequence (2024)

FAQs

Recursive Formula - Rule of Arithmetic and Geometric Sequence? ›

The recursive formula of an arithmetic sequence is, a_n = a_n-1 + d. The recursive formula of a geometric sequence is, a_n = a_n-1r.

For a geometric sequence with recurrence of the form a(n)=ra(n-1) where r is constant, each term is r times the previous term. This implies that to get from the first term to the nth term, we need to multiply by n-1 factors of r.

i.e., any term (n^th term) of an arithmetic sequence is obtained by adding the common difference (d) to its previous term ((n - 1)^th term). i.e., the recursive formula of the given arithmetic sequence is, an=an−1+d a n = a n − 1 + d .

The general form of a recursive formula for an arithmetic sequence is { a 1 = c a n = a n − 1 + d , where is the first term of the sequence, is ith term of the sequence, and is the common difference between each consecutive term of the sequence.

First, identify the common difference (how much each term in a sequence is increasing or decreasing from the previous term). State the first term of the sequence, and then write the recursive rule as (new term) = (previous term) + (common difference).

A recursive formula is a formula that defines any term of a sequence in terms of its preceding term(s). For example: The recursive formula of an arithmetic sequence is, a_n = a_n-1 + d. The recursive formula of a geometric sequence is, a_n = a_n-1r.

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by 3 gives the next term. In other words, an=a1rn−1 a n = a 1 r n - 1 .

The explicit formula for a geometric sequence is of the form a_n = a₁ [ r-1 ] , where r is the common ratio.

There are few recursive formulas to find the n^th term based on the pattern of the given data. They are, n^th term of Arithmetic Progression a_n = a_{n – 1} + d for n ≥ 2. n^th term of Geometric Progression a_n = a_{n – 1} × r for n ≥ 2.

Answer: The formula for the nth term in an arithmetic sequence is an=a1+(n−1)d. This formula can be used to determine the value of any term in an arithmetic sequence. An arithmetic sequence has a common difference between every term.

What is the difference between arithmetic and geometric sequences? ›

Arithmetic sequences are defined by an initial value and a common difference, with the same number added or subtracted to each term. Geometric sequences are defined by an initial value and a common ratio, with the same number multiplied or divided to each term.

Recursive: nth term=(n-1) term+d. Geometric: First term: a. Ratio: r. Explicit: nth term=a(r)^(n-1).

A recursive formula for a geometric sequence with common ratio r is given by an=ran–1 for n≥2. As with any recursive formula, the initial term of the sequence must be given. See Example 11.3. 3.

The recursive equation for an arithmetic squence is: f(1) = the value for the 1st term. f(n) = f(n-1) + common difference. Hope this helps.

What is the rule for the geometric sequence? Each term of a geometric sequence is formed by multiplying the previous term by a constant number r, starting from the first term a1. Therefore, the rule for the terms of a geometric sequence is an=a1(r)^(n-1).

There are two geometric probability formulas: Geometric distribution PMF: P(X = x) = (1 - p)^{x - 1}p. Geometric distribution CDF: P(X ≤ x) = 1 - (1 - p)

A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. If you know the term of an arithmetic sequence and you know the common difference , , you can find the ( n + 1 ) th term using the recursive formula a n + 1 = a n + d .

Each term of a geometric sequence is formed by multiplying the previous term by a constant number r, starting from the first term a1. Therefore, the rule for the terms of a geometric sequence is an=a1(r)^(n-1).

Expert-Verified Answer

The recursive formula for the geometric sequence 2, -10, 50, -250, ... is an = an-1 × -5, given that the first term is 2.