converting recursive sequence to explicit equation

Therefore, the next three terms should be 2 5, 2 6, and 2 7 or 32, 64, and 128.ī. It appears that each term of the sequence is a power of 2: 2 1, 2 2, 2 3, 2 4, …. Write a recursive definition for the sequence.Ī. List the next three terms of the sequence 2, 4, 8, 16, …. Therefore, for each term after the first,Ĥ=½(8), 2=½(4), 1=½(2), 0.5=½(1), 0.25=½(0.5), 0.125=½(0.25), …įor n>1, we can write the recursive definition:Īlternatively, we can write the recursive definition as:Ī rule that designates any term of a sequence can often be determined from the first few terms of the sequence.Ī. In this sequence, each term after the first is ½ the previous term. The formula that allows any term of a sequence, except the first, to be computed from the previous term is called a recursive definition.įor example, the sequence that lists the heights to which a ball bounces when dropped from a height of 16 feet is 8, 4, 2, 1, 0.5, 0.25, 0.125, …. Most sequences are sets of numbers that are related by some pattern that can be expressed as a formula. The terms of a sequence are often designated as a 1, a 2, a 3, a 4, a 5, … If the sequence is designated as the function f, then f(1)= a 1, f(2)= a 2, or in general:

In this case, the domain is the set of positive integers. Often the sequence can continue without end.

The function that lists the height of the ball after 7 bounces is shown on the graph at the right. Therefore, a sequence is a special type of function. We associate each term of the sequence with the positive integer that specifies its position in the ordered set. The general term is defined for n=1, 2, 3, 4, …. The general term or nth term of a sequence is represented by a symbol with a subscript, for example a n, t n, or u n. The dots must not be joined because n must be an integer. a pictorial or graphical representationĮarly members of a sequence can be graphed with each represented by a dot.U n= n+2 is the general term or nth term formula for n=1, 2, 3, 4, …. “The top row has three bricks and each successive row under it has one more brick than the previous row.” If u n represents the number of bricks in row n (from the top) then u 1=3, u 2=4, u 3=5, u 4=6, …. The first row has three bricks, the second row has four bricks, and the third row has five bricks. using a formula which represents the general term or nth term.Ĭonsider the illustrated tower of bricks.listing the first few terms and assuming that the pattern represented continues indefinitely.listing all terms (of a finite sequence).Sequences may be defined in one of the following ways:

Converting recursive sequence to explicit equation