1. Questions

Math

The real numbers $a$ and $b$ satisfy $|a| < 1$ and $|b| < 1.$

(a) In a grid that extends infinitely, the first row contains the numbers $1,$ $a,$ $a^2,$ $\dots.$ The second row contains the numbers $b,$ $ab,$ $a^2 b,$ $\dots.$ In general, each number is multiplied by $a$ to give the number to the right of it, and each number is multiplied by $b$ to give the number below it.

Find the sum of all numbers in the grid.

[asy]
unitsize(1 cm);

int i, j;

for (i = 0; i <= 4; ++i) {
draw((i,0)--(i,-4.5));
draw((0,-i)--(4.5,-i));
}

label("$1$", (0.5,-0.5));
label("$a$", (1.5,-0.5));
label("$a^2$", (2.5,-0.5));
label("$a^3$", (3.5,-0.5));

label("$b$", (0.5,-1.5));
label("$ab$", (1.5,-1.5));
label("$a^2 b$", (2.5,-1.5));
label("$a^3 b$", (3.5,-1.5));

label("$b^2$", (0.5,-2.5));
label("$ab^2$", (1.5,-2.5));
label("$a^2 b^2$", (2.5,-2.5));
label("$a^3 b^2$", (3.5,-2.5));

label("$b^3$", (0.5,-3.5));
label("$ab^3$", (1.5,-3.5));
label("$a^2 b^3$", (2.5,-3.5));
label("$a^3 b^3$", (3.5,-3.5));

label("$\dots$", (5,-2));
label("$\vdots$", (2,-5));
[/asy]

(b) Now suppose the grid is colored like a chessboard, with alternating black and white squares, as shown below. Find the sum of all the numbers that lie on the black squares.

[asy]
unitsize(1 cm);

int i, j;

for (i = 0; i <= 3; ++i) {
for (j = 0; j <= 3; ++j) {
if ((i + j) % 2 == 0) {
fill(shift((i,-j))*((0,0)--(1,0)--(1,-1)--(0,-1)--cycle),black);
}
}}

fill((0,-4)--(1,-4)--(1,-4.5)--(0,-4.5)--cycle,black);
fill((2,-4)--(3,-4)--(3,-4.5)--(2,-4.5)--cycle,black);
fill((4,0)--(4,-1)--(4.5,-1)--(4.5,0)--cycle,black);
fill((4,-2)--(4,-3)--(4.5,-3)--(4.5,-2)--cycle,black);
fill((4,-4)--(4.5,-4)--(4.5,-4.5)--(4,-4.5)--cycle,black);

for (i = 0; i <= 4; ++i) {
draw((i,0)--(i,-4.5));
draw((0,-i)--(4.5,-i));
}

label("$1$", (0.5,-0.5), white);
label("$a$", (1.5,-0.5));
label("$a^2$", (2.5,-0.5), white);
label("$a^3$", (3.5,-0.5));

label("$b$", (0.5,-1.5));
label("$ab$", (1.5,-1.5), white);
label("$a^2 b$", (2.5,-1.5));
label("$a^3 b$", (3.5,-1.5), white);

label("$b^2$", (0.5,-2.5), white);
label("$ab^2$", (1.5,-2.5));
label("$a^2 b^2$", (2.5,-2.5), white);
label("$a^3 b^2$", (3.5,-2.5));

label("$b^3$", (0.5,-3.5));
label("$ab^3$", (1.5,-3.5), white);
label("$a^2 b^3$", (2.5,-3.5));
label("$a^3 b^3$", (3.5,-3.5), white);

label("$\dots$", (5,-2));
label("$\vdots$", (2,-5));
[/asy]

Sorry but I don't know how to show the grids.

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Anonymous

  1. each row is a geometric progression. Starting with n=0, row n is the GP
    b^n (1 + a + a^2 + ...)
    so the sum Sn of row n is
    b^n * 1/(1-a)
    Now you have another GP, which is the sum of all the row sums. That is
    1/(1-a) (1 + b + b^2 + ...) whose sum is 1/(1-a) * 1/(1-b)

    So the grand total of the whole grid is 1/((1-a)(1-b))

    1. 👍
    2. 👎
    oobleck

  2. wertwetrwetw

    1. 👍
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    rterwt

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