Consider the expansion of eight 𝑥 plus two 𝑦 to the power of 23. Find the ratio between the eighth and the seventh terms.
In order to actually find the answer to this and actually find the ratio between the eighth and the seventh terms, what we’re actually gonna do is actually find the ratio between consecutive terms in a binomial expansion. Now in order to find this ratio, what we actually have is a formula that we can use. And the formula tells us that if we have a binomial expansion in the form 𝑎 plus 𝑏 to the power of 𝑛, then the ratio of two consecutive terms is equal to 𝑛 minus 𝑟 plus one over 𝑟 multiplied by 𝑏 over 𝑎.
Okay, great, so we now have a formula for actually finding the ratio between two consecutive terms. So what we need to do now is actually use this to find the ratio between our eighth and seventh terms. So what we have is the ratio between the eighth and seventh terms is equal to and now here is the point but we have to actually work out so what are our 𝑟 and what are our 𝑛.
Well, our 𝑟 is equal to seven and our 𝑛 is equal to 23 because this is the exponent that the actual parenthesis raise to. And then, 𝑎 is equal to eight 𝑥 because that’s our first term and 𝑏 is equal to two 𝑦 because this is our second time. Okay, great, so we’ve now got all our unknowns, we can actually put them into our formula and find the ratio of our terms. So we get that the ratio between the eighth and seventh terms is equal to 23 — because that’s our 𝑛 — minus seven — our 𝑟 — plus one all divided by seven because again that’s our 𝑟. And then, this is multiplied by 𝑏 over 𝑎 — so two 𝑦 over eight 𝑥.
Okay, great, so now, let’s start to simplify. So this gives us 17 over seven multiplied by two 𝑦 over eight 𝑥. So now as we’re multiplying fractions, we just multiply the numerators and denominators, which gives us 34𝑦 over 56𝑥. So then, finally, we divide the numerator and denominator by two because that’s a factor of both 34 and 56.
So therefore, we can say that the ratio between the eighth and the seventh terms is equal to 17𝑦 over 28𝑥.