general term

(noun)

A mathematical expression containing variables and constants that, when substituting integer values for each variable, produces a valid term in a sequence.

Related Terms

Examples of general term in the following topics:

  • The General Term of a Sequence

    • Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.
    • Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence.
    • Some sequences are generated by a general term which is not a polynomial.
    • Given any general term, the sequence can be generated by plugging in successive values of $n$.
    • Practice finding a formula for the general term of a sequence

  • Sums, Differences, Products, and Quotients

    • A monomial equations has one term; a binomial has two terms; a trinomial has three terms.
    • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
    • Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
    • The general form is shown in and is diagrammed in .
    • The general form of the FOIL method using only variables as the potential multipliers.

  • Recursive Definitions

    • When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way:
    • The above equation is an example of a recursive equation since the $n$th term can only be calculated by considering the previous term in the sequence.
    • In this equation, one can directly calculate the nth-term of the arithmetic sequence without knowing the previous terms.
    • One can work out every term in the series just by knowing previous terms.
    • As can be seen from the examples above, working out and using the previous term $a_{n−1}$ can be a much simpler computation than working out $a_{n}$ from scratch using a general formula.

  • Multiplying Algebraic Expressions

    • A polynomial is called a binomial if it has two terms, and a trinomial if it has three terms.
    • Any negative sign on a term should be included in the multiplication of that term.
    • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
    • Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
    • Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.

  • Introduction to Sequences

    • A geometric sequence is a list in which each number is generated by multiplying a constant by the previous number.
    • The first term is always $t_1$.
    • From this you can see the generalization that:
    • The first term is $t_1$; the second term is $r$ times that, or $t_1r$; the third term is $r$ times that, or $t_1r^2$; and so on.
    • So the general rule is:

  • Sums and Series

    • The notation for this operation is to use the capital greek letter sigma, $\Sigma$, following the general formula:
    • You understand that this trick will not work, in general, for series that are not arithmetic.
    • If we apply this trick to the generic arithmetic series, we get a formula that can be used to sum up any arithmetic series.
    • Finally—once again—we can apply this trick to the generic geometric series to find a formula.
    • So there we have it: a general formula for the sum of any finite geometric series, with the first term $t_1$ , the common ratio $r$ , and a total of $n$ terms.

  • Summing the First n Terms in a Geometric Sequence

    • By utilizing the common ratio and the first term of a geometric sequence, we can sum its terms.
    • The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant.
    • is geometric, because each successive term can be obtained by multiplying the previous term by $\displaystyle{\frac{1}{2}}$.
    • Therefore, by utilizing the common ratio and the first term of the sequence, we can sum the first $n$ terms.
    • Also, note that $r = 3$, because each term is multiplied by a factor of $3$ to find the subsequent term.

  • Geometric Sequences

    • The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by
    • Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
    • Positive, the terms will all be the same sign as the initial term
    • Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.
    • Calculate the $n$th term of a geometric sequence given the initial value $a$ and common ratio $r$

  • Summing Terms in an Arithmetic Sequence

    • First we think of it as the sum of terms that are written in terms of $a_1$, so that the second term is $a_1+d$, the third is $a_1+2d$, and so on.
    • Next, we think of each term as being written in terms of the last term, $a_n$.
    • Then the last term is $a_n$, the term before the last is $a_n-d$, the term before that is $a_n-2d$, and so on.
    • We can see that the first term is $a_1 = 3$.
    • The general form for an infinite arithmetic series is:

  • Applications and Problem-Solving

    • Adding the first term to the last term: $3 + 17 = 20$
    • Adding the second term to the second-to-last term also amounts to a sum of 20.
    • We can see that the third term and third-to-last terms have a similar effect.
    • There are eight terms in $3+5+7+9+11+13+15+17$, and they add to four 20s, or 80.
    • Adding the first and the last, second term and second to last, etc. all yield the same answer.