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.

  1. 👍
  2. 👎
  3. 👁
  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. 👎
  2. wertwetrwetw

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. alebra

    Let f(x)=2x^2+x-3 and g(x)=x-1. Perform the indicated operation then find the domain. (f*g)(x) a.2x^3-x^2-4x+3; domain:all real numbers b.2x^3+x^2-3x; domain: all real numbers c.2x^3+x^2+4x-3; domain: negative real numbers

  2. Precalc

    To which set(s) of numbers does the number sqrt -16 belong? Select all that apply. real numbers complex numbers*** rational numbers imaginary numbers*** irrational numbers I can only pick two, and that's what I think it is. Please

  3. Pre-Calculus

    A rectangle is bounded by the x-axis and the semicircle y = √36 – x2, as shown in the figure below. Write the area A of the rectangle as a function of x, and determine the domain of the area function. A = all real numbers

  4. Algebra II

    Given no other restrictions, what are the domain and range of the following function? f(x)=x^2-2x+2 A. D= all real numbers R={y|y>=1} B. R=all real numbers D={x|x>=1} I think it is A...?

  1. algebra

    Let $x$, $y$, and $z$ be positive real numbers that satisfy \[2 \log_x (2y) = 2 \log_{2x} (4z) = \log_{2x^4} (8yz) \neq 0.\] The value of $xy^5 z$ can be expressed in the form $\frac{1}{2^{p/q}}$, where $p$ and $q$ are relatively

  2. Algebra

    To which subset of real numbers does the following number belong? square root of 7 A)rational numbers B)irrational numbers****** C)whole numbers, integers, rational numbers D)whole numbers, natural numbers, integers

  3. Math

    1. The sum of a number and 2 is 6 less than twice that number. 2. A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is 20 feet. 4. Six times a number is less than 72.

  4. Math

    Which set of best describes the numbers used on a scale for a standard thermometer? A.whole numbers B.rational numbers C.real numbers D.integers I choose Real Numbers if I am incorrect can you explain why.

  1. Algebra

    Choose which group of sets the following number belongs to. Be sure to account for ALL sets. 2/5 A. Real numbers, rational numbers. B. Real numbers, irrational numbers. C. Rational numbers, natural numbers, integers. D. Irrational

  2. AiMS MATH

    -5 Is A Member Of Which Subset Of Real Numbers. a) natural numbers b) whole numbers c) intergers d) rational numbers e) irrational numbers

  3. Math

    The twice–differentiable function f is defined for all real numbers and satisfies the following conditions: f(0)=3 f′(0)=5 f″(0)=7 a)The function g is given by g(x)=e^ax+f(x) for all real numbers, where a is a constant. Find

  4. Calculus

    Let g and h be any two twice-differentiable functions that are defined for all real numbers and that satisfy the following properties for all x: I) (g(x))^2 + (h(x))^2=1 ii) g'(x)= (h(x))^2 iii) h(x)>0 iv) g(0)=0 a)Justify that

You can view more similar questions or ask a new question.