anation":{"methods":[{"id":"factoring_standard","label":"Factoring to Standard Form","steps":[{"type":"text","order":0,"title":"Understand the validity condition","content":"For a binomial expansion of the form $(1 + X)^n$, where $n$ is not a positive integer, the series converges and is valid if and only if the absolute value of the variable term is less than $1$:\n\n$$|X| < 1$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$G(\\omega) = (\\htmlData{anchor=orig}{25} + 2\\omega)^{-\\frac{1}{2}}$"},{"latex":"$$G(\\omega) = \\htmlData{anchor=factored-out}{25^{-\\frac{1}{2}}} \\left(1 + \\htmlData{anchor=standard-x}{\\color{@sky}{\\frac{2}{25}\\omega}}\\right)^{-\\frac{1}{2}}$"}],"highlights":[{"color":"amber","style":"box","anchor":"orig"},{"color":"sky","style":"text-color","anchor":"standard-x"}],"connections":[{"to":"factored-out","from":"orig","color":"amber","label":"factor out 25","style":"solid"}]},"order":1,"title":"Factor to standard form","description":"To apply the validity condition, we first rewrite the expression by factoring out 25 so it is in the standard form $(1 + X)^n$."},{"type":"text","order":2,"title":"Apply the validity condition","content":"Comparing our factored expression to $(1 + X)^n$, we identify:\n\n$$X = \\frac{2}{25}\\omega$$\n\nSubstituting this into the convergence condition $|X| < 1$ gives:\n\n$$\\left|\\frac{2}{25}\\omega\\right| < 1$$"},{"type":"text","order":3,"title":"Solve the inequality","content":"We solve the inequality for $|\\omega|$ by multiplying both sides by $\\frac{25}{2}$:\n\n$$|\\omega| < \\frac{25}{2}$$","highlights":[{"text":"|\\omega| < \\frac{25}{2}","index":1,"location":"option"}]}]},{"id":"shortcut_formula","label":"Direct Formula Application","steps":[{"type":"text","order":0,"title":"Identify the validity formula","content":"To find the range of validity for a general binomial expansion of the form $(a + bx)^n$, where $n$ is not a positive integer, we use the standard validity condition:\n\n$$\\left| \\frac{bx}{a} \\right| < 1$$\n\nThis can be rearranged into a direct formula for the variable $x$:\n\n$$|x| < \\left| \\frac{a}{b} \\right|$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Comparing } (\\htmlData{anchor=a-term}{\\color{@sky}{25}} + \\htmlData{anchor=b-term}{\\color{@amber}{2}}\\htmlData{anchor=x-term}{\\color{@purple}{\\omega}})^{-\\frac{1}{2}} \\text{ to } (a + bx)^n$"},{"latex":"$$\\text{We get: } \\htmlData{anchor=a-val}{\\color{@sky}{a = 25}}, \\quad \\htmlData{anchor=b-val}{\\color{@amber}{b = 2}}, \\quad \\htmlData{anchor=x-val}{\\color{@purple}{x = \\omega}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"b-term"},{"color":"purple","style":"text-color","anchor":"x-term"}],"connections":[{"to":"a-val","from":"a-term","color":"sky","label":"value of a","style":"solid"},{"to":"b-val","from":"b-term","color":"amber","label":"value of b","style":"solid"},{"to":"x-val","from":"x-term","color":"purple","label":"variable x","style":"solid"}]},"order":1,"title":"Substitute the given values","description":"We identify the values of $a$, $b$, and the variable $x$ by comparing the expression $(25 + 2\\omega)^{-\\frac{1}{2}}$ directly with $(a + bx)^n$."},{"type":"text","order":2,"title":"Calculate the validity range","content":"Now we substitute these parameters into our direct formula:\n\n$$\\begin{aligned} |\\omega| &< \\left| \\frac{25}{2} \\right| \\\\ |\\omega| &< \\frac{25}{2} \\end{aligned}$$\n\nThis is the range of values for which the expansion is valid.","highlights":[{"text":"|\\omega| < \\frac{25}{2}","index":1,"location":"option"}]}]}],"generatedAt":"2026-06-09T04:35:21.525Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"2ad0e20c-a359-4c04-bf1f-86a5931d2695","label":"b","content":"Find the value of \$a\$.\n\nChoose from the options below:\n\\[\n\\frac{1}{25} \\quad \\frac{1}{5} \\quad 5 \\quad 25\n\\]","markscheme":"- Evaluates the leading constant term as $a = 25^{-\\frac{1}{2}} = \\frac{1}{5}$ B1"

Binomial Expansion

Question 1

Binomial Expansion Questions

Binomial Expansion

Question 1

Binomial Expansion Questions