Question

Linear recurrence relation

Answer 1

The query pertains to 'linear recurrence relations' in Mathematics, particularly concerning high school algebra and series expansions, such as the binomial theorem, and plotting data on a logarithmic scale.

Step-by-step explanation:

The term linear recurrence relation refers to a sequence of numbers where each term is a linear combination of previous terms. The relationship is defined by two aspects: the number of terms that go into the combination (order of the relation), and the coefficients of each of those terms. A classic example of a linear recurrence relation is the Fibonacci sequence, where each term is the sum of the two preceding terms.

When addressing series expansions like the binomial theorem, it is an expression that allows us to expand polynomials raised to a power in a series format. The theorem is a key concept in algebra and is particularly useful for calculating powers of binomials and deriving coefficients of individual terms within expanded polynomials.

To plot data like the recurrence interval on a logarithmic scale, it is essential to understand that each increment on the axis represents a multiplication by a certain factor, rather than a linear addition. This type of representation is particularly useful when dealing with data that varies by orders of magnitude.

Answer 2

True

A linear recurrence relation involving a sequence of numbers
`a_n` is one of the form

`\displaystyle\sum_(k=0)^nc_(n-k)a_(n-k)=c_na_n+c_(n-1)a_(n-1)+\cdots+c_2a_2+c_1a_1=c`

where
`c_1,c_2,\ldots,c_n` and
`c` are any fixed numbers.

The given recurrence can be rearranged as

`a_n=a_(n-1)+2\implies 1\cdot a_n+(-1)\cdot a_(n-1)=2`

A nonlinear recurrence would have a more "exotic" form that cannot be written in the form above. Some example:

`a_n+\frac1{a_(n-1)}=1`

`a_na_(n-1)=\pi`

`{a_n}^2+\sqrt{a_(n-1)}-\left((a_(n-2))/(√(a_n))\right)^{a_(n-3)}=0`