Question

use pascals triangle to complete the expansion of $(t + u)^7$. $t^7 + 7t^6usquare + 21t^square u^2 + 35t^4u^3 + 35t^3u^4 + 21t^2u^5 + 7tu^6 + u^7$

Explanation:

Step1: Recall Binomial Theorem

The binomial expansion of \((a + b)^n\) is given by \(\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), where \(\binom{n}{k}\) are the binomial coefficients (from Pascal's Triangle for \(n\)th row). For \((t + u)^7\), the general term is \(\binom{7}{k}t^{7 - k}u^{k}\).

Step2: Analyze the first blank (exponent of \(u\) in \(7t^{6}u^{\square}\))

For the term with \(t^{6}\), we have \(n - k=6\) (since \(n = 7\)), so \(7 - k=6\) implies \(k = 1\). So the exponent of \(u\) should be \(1\).

Step3: Analyze the second blank (exponent of \(t\) in \(21t^{\square}u^{2}\))

For the term with \(u^{2}\), we have \(k = 2\). Then \(n - k=7 - 2 = 5\). So the exponent of \(t\) should be \(5\).

Answer:

First blank: \(1\), Second blank: \(5\)

So the completed terms are \(7t^{6}u^{1}\) and \(21t^{5}u^{2}\)