Example Of An Arithmetic Sequence

Arithmetic sequence

An arithmetics sequence is a type of sequence in which the departure between each consecutive term in the sequence is constant. For case, the difference between each term in the following sequence is 3:

2, 5, 8, xi, 14, 17, xx...

To expand the higher up arithmetics sequence, starting at the first term, 2, add 3 to determine each sequent term. This is elementary for the first few terms, but using this method to make up one's mind terms further along in the sequence gets tedious very quickly. Fortunately, the nthursday term of an arithmetic sequence can be determined using

a_n = a₁ + (n - 1)d

where a_n is the n ^th term, a₁ is the initial term, and d is the abiding difference betwixt each term. Using the above sequence, the formula becomes:

a_n = 2 + 3n - 3 = 3n - 1

Therefore, the 100th term of this sequence is:

a₁₀₀ = 3(100) - i = 299

This formula allows usa to decide the due northth term of whatsoever arithmetic sequence.

Arithmetic sequence vs arithmetic series

An arithmetic series is the sum of a finite function of an arithmetic sequence. For example, 2 + 5 + eight = xv is an arithmetic serial of the first three terms in the sequence above. The sum of a finite arithmetic sequence can be plant using the following formula,

where n is the number of terms in the sequence, a₁ is the first term in the sequence, and a_n is the n ^th term, and d is the abiding difference between each term.

Instance

Find the sum of the starting time 7 terms in the arithmetic sequence 1, 7, 13, 19, 25, ...

First decide the rest of the first vii terms. The difference between each consecutive term in the sequence is 6. Since nosotros merely demand to determine ii more than terms, it is simpler to just add vi to the last given term and the term after that. The kickoff 7 terms are:

ane, 7, 13, nineteen, 25, 31, 37

The sum of these terms is: