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MATH / Consider this pattern where the positive, proper fractions with denominator…

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Answer published by the source. Consult the official source to check your work against its answer.

problem

Consider this pattern where the positive, proper fractions with denominator (n+1)(n+1) are arranged in the nnth row in a triangular formation. The 1st through 4th rows are shown; each row has one more entry than the previous row. What is the sum of the fractions in the 15th row? [asy] label("12\frac{1}{2}",(0,0),S); label("13\frac{1}{3}",(-5,-5),S); label("23\frac{2}{3}",(5,-5),S); label("14\frac{1}{4}",(-10,-10),S); label("24\frac{2}{4}",(0,-10),S); label("34\frac{3}{4}",(10,-10),S); label("15\frac{1}{5}",(-15,-15),S); label("25\frac{2}{5}",(-5,-15),S); label("35\frac{3}{5}",(5,-15),S); label("45\frac{4}{5}",(15,-15),S); dot((0,-22)); dot((0,-20)); dot((0,-24)); [/asy]
Plain-text mathematical notation (without MathML)
Consider this pattern where the positive, proper fractions with denominator (n+1)  are arranged in the nth row in a triangular formation. The 1st through 4th rows are shown; each row has one more entry than the previous row. What is the sum of the fractions in the 15th row?

[asy]
label("(1)/(2)",(0,0),S);
label("(1)/(3)",(-5,-5),S);
label("(2)/(3)",(5,-5),S);
label("(1)/(4)",(-10,-10),S);
label("(2)/(4)",(0,-10),S);
label("(3)/(4)",(10,-10),S);
label("(1)/(5)",(-15,-15),S);
label("(2)/(5)",(-5,-15),S);
label("(3)/(5)",(5,-15),S);
label("(4)/(5)",(15,-15),S);
dot((0,-22));
dot((0,-20));
dot((0,-24));
[/asy]
Original LaTeX notation
Consider this pattern where the positive, proper fractions with denominator $(n+1)$  are arranged in the $n$th row in a triangular formation. The 1st through 4th rows are shown; each row has one more entry than the previous row. What is the sum of the fractions in the 15th row?

[asy]
label("$\frac{1}{2}$",(0,0),S);
label("$\frac{1}{3}$",(-5,-5),S);
label("$\frac{2}{3}$",(5,-5),S);
label("$\frac{1}{4}$",(-10,-10),S);
label("$\frac{2}{4}$",(0,-10),S);
label("$\frac{3}{4}$",(10,-10),S);
label("$\frac{1}{5}$",(-15,-15),S);
label("$\frac{2}{5}$",(-5,-15),S);
label("$\frac{3}{5}$",(5,-15),S);
label("$\frac{4}{5}$",(15,-15),S);
dot((0,-22));
dot((0,-20));
dot((0,-24));
[/asy]

level

Level 4

type

Algebra

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Source and history

task_id
f0a25e9d-d474-57f5-847a-c237adc047f7
task_key
algebra--test--f0a25e9d-d474-57f5-847a-c237adc047f7
task_revision_id
3
config / split
algebra / test

Official source

initial import