Problems

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Let the sequences of numbers {an} and {bn}, that are associated with the relation Δbn=an (n=1,2,), be given. How are the partial sums Sn of the sequence {an} Sn=a1+a2++an linked to the sequence {bn}?

Definition. Let the function f(x,y) be valid at all points of a plane with integer coordinates. We call a function f(x,y) harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: f(x,y)=1/4(f(x+1,y)+f(x1,y)+f(x,y+1)+f(x,y1)). Let f(x,y) and g(x,y) be harmonic functions. Prove that for any a and b the function af(x,y)+bg(x,y) is also harmonic.

Definition. The sequence of numbers a0,a1,,an,, which, with the given p and q, satisfies the relation an+2=pan+1+qan (n=0,1,2,) is called a linear recurrent sequence of the second order.

The equation x2pxq=0 is called a characteristic equation of the sequence {an}.

Prove that, if the numbers a0, a1 are fixed, then all of the other terms of the sequence {an} are uniquely determined.

The following words/sounds are given: look, yar, yell, lean, lease. Determine what will happen if the sounds that make up these words are pronounced in reverse order.

In the entry +++++++= replace the asterisks with different digits so that the equality is correct.