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MATH / Let f(m,1)=f(1,n)=1 for m≥1,n≥1, and let f(m,n)=f(m−1,n)+f(m,n−1)+f(m−1,n−1) for m>1 and n>1. Also, let S(k)=∑_(a+b=k)f(a,b), for a≥1,b≥1.Note: The summation notation means to sum …

Problem

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problem

Let f(m,1)=f(1,n)=1f(m,1) = f(1,n) = 1 for m≥1,n≥1,m \geq 1, n \geq 1, and let f(m,n)=f(m−1,n)+f(m,n−1)+f(m−1,n−1)f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1) for m>1m > 1 and n>1.n > 1. Also, let S(k)=∑a+b=kf(a,b), for a≥1,b≥1.S(k) = \sum_{a+b=k} f(a,b), \text{ for } a \geq 1, b \geq 1.Note: The summation notation means to sum over all positive integers a,ba,b such that a+b=k.a+b=k. Given that S(k+2)=pS(k+1)+qS(k) for all k≥2,S(k+2) = pS(k+1) + qS(k) \text{ for all } k \geq 2,for some constants pp and qq, find pq.pq.
Plain-text mathematical notation (without MathML)
Let f(m,1)=f(1,n)=1 for m≥1,n≥1, and let f(m,n)=f(m−1,n)+f(m,n−1)+f(m−1,n−1) for m>1 and n>1.  Also, let

S(k)=∑_(a+b=k)f(a,b), for a≥1,b≥1.Note: The summation notation means to sum over all positive integers a,b such that a+b=k.

Given that

S(k+2)=pS(k+1)+qS(k) for all k≥2,for some constants p and q, find pq.
Original LaTeX notation
Let $f(m,1) = f(1,n) = 1$ for $m \geq 1, n \geq 1,$ and let $f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1)$ for $m > 1$ and $n > 1.$  Also, let

$$S(k) = \sum_{a+b=k} f(a,b), \text{ for } a \geq 1, b \geq 1.$$Note: The summation notation means to sum over all positive integers $a,b$ such that $a+b=k.$

Given that

$$S(k+2) = pS(k+1) + qS(k) \text{ for all } k \geq 2,$$for some constants $p$ and $q$, find $pq.$

level

Level 3

type

Intermediate Algebra

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Source and history

task_id
963c3834-9814-5802-a741-e09b784400c1
task_key
intermediate~5falgebra--test--963c3834-9814-5802-a741-e09b784400c1
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import