Gerard M. answered • 12/08/20
Enthusiastic Math Tutor With 4 Years Experience
Hi Kush,
The first step is to find the pattern in the sequence. The most common patterns are simply adding by a number repeatedly (arithmetic sequence) or multiplying by a number repeatedly (geometric sequence). This pattern seems not to be arithmetic, but geometric, and we can make sure by dividing each term by the previous term: -27 ÷ -9 = 3, -9 ÷ -3 = 3, -3 ÷ -1 = 3. See how the ratio between the terms is always 3? That means we're multiplying by 3 to get the next term.
Now that we know the pattern, let's try to formalize it for any nth term in the sequence, an. The 1st term, a1, is -1. Then, to get to the next term, a2, we multiply by 3, -1 × 3. If we continue, the pattern starts to look like this:
Let's break down that last formula. We see that for each term we're multiplying by another 3, and you might recognize repeated multiplication as exponents (3 × 3 = 32 and 3 × 3 × 3 = 33). The ^ is the symbol for exponents, in case you didn't know (32 = 3^2). And notice, the number of times we repeat ×3 for each term is 1 less than the term number n (a3 repeats ×3, 2 times). Hence, we raise 3 to the (n - 1) power, multiplied by -1 because that's our first term.
Now that we have our formula, an = -1 × 3n-1, we can write a12 = -1 × 312-1 = -1 × 311. If you're allowed to use a calculator, you can type that in and get your answer, but if not, you probably don't want to do out the whole calculation. So let's see if we can estimate and find the most reasonable answer out of the four given.
311 just means 3 multiplied 11 times: 3×3×3×3×3×3×3×3×3×3×3. Notice, we can group these 3's however we like, for example: (3×3)×(3×3)×(3×3)×(3×3)×(3×3)×3 which is the same as 9×9×9×9×9×3. If we want to use exponents, we could also write this as 3 × 95. Remember, we want to estimate, so how can we modify this to an answer that's somewhat close, but easier to calculate? Well 9 is close to 10, and 105 is easy to calculate, so let's try that. That gives us 3 × 105 = 300,000. Now let's review our options.
The closest two options in magnitude are a) 118,098 and d) -177,147, which are both 6-digit numbers, like 300,000. However, the answer has to be negative, since all of the terms are negative, so a) is crossed out. Which leaves d) as our most likely option, since b) and c) are 5-digits.
If this is good enough for you, then you have your answer. But if you're not convinced, or you'd like to really narrow down your answer, let me know and I'll demonstrate it. But give it a try first! You never know what you might come up with :)
