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n200080 [17]
3 years ago
8

MNOP is a trapezoid with median QR. Find x

Mathematics
1 answer:
nikitadnepr [17]3 years ago
5 0

\bf  \large \rightarrow \: \:2x \:  +  \: 8 \:  =  \: 0

\bf  \large \rightarrow \: \:x \:  =  \:  \frac{8}{2}  \\

\bf  \large \rightarrow \: \:x \:  =  \:    \cancel\frac{ 8}{ 2} \:   \: ^{4}   \\

\bf  \large \rightarrow \: \:x \:  =  \:  4

Option ( A ) is the correct answer.

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Prove\ that\ the\ assumption \is \true for\ n=1\\1^3=\frac{1^2(1+1)^2}{4}\\ 1=\frac{4}{4}=1\\

Formula works when n=1

Assume the formula also works, when n=k.

Prove that the formula works, when n=k+1

1^3+2^3+3^3...+k^3+(k+1)^3=\frac{(k+1)^2(k+2)^2}{4} \\\frac{k^2(k+1)^2}{4}+(k+1)^3=\frac{(k+1)^2(k+2)^2}{4} \\\frac{k^2(k^2+2k+1)}{4}+(k+1)^3=\frac{(k^2+2k+1)(k^2+4k+4)}{4} \\\frac{k^4+2k^3+k^2}{4}+k^3+3k^2+3k+1=\frac{k^4+4k^3+4k^2+2k^3+8k^2+8k+k^2+4k+4}{4}\\\\\frac{k^4+2k^3+k^2}{4}+k^3+3k^2+3k+1=\frac{k^4+6k^3+13k^2+12k+4}{4}\\\frac{k^4+2k^3+k^2}{4}+\frac{4k^3+12k^2+12k+4}{4}=\frac{k^4+6k^3+13k^2+12k+4}{4}\\\frac{k^4+6k^3+13k^2+12k+4}{4}=\frac{k^4+6k^3+13k^2+12k+4}{4}\\

Since the formula has been proven with n=1 and n=k+1, it is true. \square

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