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The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$? | Since $18 - 14 = 4$, the common difference in the first column of squares is 4, so the number above 14 is $14 - 4 = 10$, and the number above 10 is $10 - 4 = 6$. This is also the fourth number in the row, so the common difference in the row is $(6 - 21)/3 = -5$.
Then the seventh (and last) number in the row is $21 - 5 \cdot 6 = -9$. In the second column, the common difference is $[(-17) - (-9)]/4 = -2$, so $N = -9 - (-2) = \boxed{-7}$. | Since $18 - 14 = 4$, the common difference in the first column of squares is 4, so the number above 14 is $14 - 4 = 10$, and the number above 10 is $10 - 4 = 6$. This is also the fourth number in the row, so the common difference in the row is $(6 - 21)/3 = -5$.
Then the seventh (and last) number in the row is $21 - 5 \cdot 6 = -9$. In the second column, the common difference is $[(-17) - (-9)]/4 = -2$, so $N = -9 - (-2) = \boxed{-7}$. | algebra_1010 | 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"Below is a portion of the graph of a function, $y=u(x)$:What is the exact value of $u(-2.33)+u(-0.8(...TRUNCATED) | "We can't read off the exact value of $u(-2.33)$ or $u(-0.81)$ or $u(0.81)$ or $u(2.33)$ from the gr(...TRUNCATED) | "We can't read off the exact value of $u(-2.33)$ or $u(-0.81)$ or $u(0.81)$ or $u(2.33)$ from the gr(...TRUNCATED) | algebra_1076 | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) | Not supported with pagination yet | ||
"Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$(...TRUNCATED) | "By the distance formula, \\begin{align*}\nAP &= \\sqrt{(4-0)^2 + (2-0)^2} = \\sqrt{16 + 4} = 2\\sq(...TRUNCATED) | "By the distance formula, \\begin{align*}\nAP &= \\sqrt{(4-0)^2 + (2-0)^2} = \\sqrt{16 + 4} = 2\\sq(...TRUNCATED) | algebra_1091 | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) | Not supported with pagination yet | ||
The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a-b+c$. | "When $x=-1$, we have $y = a-b+c$. The graph appears to pass through $(-1,-2)$. Since $a$, $b$, an(...TRUNCATED) | "When $x=-1$, we have $y = a-b+c$. The graph appears to pass through $(-1,-2)$. Since $a$, $b$, an(...TRUNCATED) | algebra_1119 | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) | Not supported with pagination yet | ||
"Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse(...TRUNCATED) | "For $f$ to have an inverse function, it must not take any repeated value -- that is, we must not ha(...TRUNCATED) | "For $f$ to have an inverse function, it must not take any repeated value -- that is, we must not ha(...TRUNCATED) | algebra_1536 | Not supported with pagination yet | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) |
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"A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.\n\nWhat is the value of $8a-4b+2c-d(...TRUNCATED) | "Note that $f(-2) = a(-8)+b(4)+c(-2)+d$. Therefore, $$8a-4b+2c-d = -f(-2).$$Since the point $(-2,-3)(...TRUNCATED) | "Note that $f(-2) = a(-8)+b(4)+c(-2)+d$. Therefore, $$8a-4b+2c-d = -f(-2).$$Since the point $(-2,-3)(...TRUNCATED) | algebra_1562 | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) | Not supported with pagination yet | ||
"Let $f$ be defined by \\[f(x) = \\left\\{\n\\begin{array}{cl}\n2-x & \\text{ if } x \\leq 1, \\\\\(...TRUNCATED) | "The number $f^{-1}(-3)$ is the value of $x$ such that $f(x) = -3$. Since the function $f$ is defin(...TRUNCATED) | "The number $f^{-1}(-3)$ is the value of $x$ such that $f(x) = -3$. Since the function $f$ is defin(...TRUNCATED) | algebra_1614 | Not supported with pagination yet | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) |
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"The graph of the equation $y = \\frac{x}{x^3 + Ax^2 + Bx + C}$, where $A,B,C$ are integers, is show(...TRUNCATED) | "We can see from the graph that there are three vertical asymptotes at $x = -2, 1, 2$. It follows th(...TRUNCATED) | "We can see from the graph that there are three vertical asymptotes at $x = -2, 1, 2$. It follows th(...TRUNCATED) | algebra_1631 | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) | Not supported with pagination yet | ||
"A triangular region is bounded by the two coordinate axes and the line given by the equation $2x + (...TRUNCATED) | "To start, use the equation to solve for the $x$ and $y$ intercepts of the line. Letting $x$ equal (...TRUNCATED) | "To start, use the equation to solve for the $x$ and $y$ intercepts of the line. Letting $x$ equal (...TRUNCATED) | algebra_1653 | Not supported with pagination yet | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) |
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"Let $f(x)$ be the function defined on $-1\\le x\\le 1$ by the formula $$f(x)=1-\\sqrt{1-x^2}.$$This(...TRUNCATED) | "The graph of $x=f(y)$ can be drawn by reflecting the graph of $y=f(x)$ across the line $y=x$:The en(...TRUNCATED) | "The graph of $x=f(y)$ can be drawn by reflecting the graph of $y=f(x)$ across the line $y=x$:The en(...TRUNCATED) | algebra_1654 | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) | "/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAIBAQEBAQIBAQECAgICAgQDAgICAgUEBAMEBgUGBgYFBgYGBwkIBgcJBwYGCAsICQo(...TRUNCATED) |
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