Find the first term of the sequence 5,15,45,135 that exceeds 20,000
a = 5
r = 3
5*3^n > 20000
3^n > 4000
n > log4000/log3 = 9.73
So the 9 term will exceed 20,000
check:
a8 = 5*3^7 = 10935
a9 = 5*3^8 = 32805
9term
Oh boy, let me put on my math hat and do some calculations here... Okay, so let's find the common ratio by dividing each term by the one before it. Doing that, we get 3, 3, and 3. Which means the sequence is multiplying by 3 each time. Now, to find the first term that exceeds 20,000, we can keep multiplying the last term in the sequence (135) by 3 until we hit the jackpot. After some number crunching, I can confidently say that the first term to exceed 20,000 is... (drumroll please)... 72945! So there you have it, the first term of the sequence 5, 15, 45, 135 that exceeds 20,000 is 72945. Now that's a big number, isn't it?
To find the first term of the sequence that exceeds 20,000, we can see that the sequence is formed by multiplying each term by 3.
Let's set up an equation to find the value of the first term (a) that exceeds 20,000:
a * 3^n > 20,000
Here, n represents the number of terms in the sequence.
To solve for n, we can use the equation:
3^n > 20,000 / a
Taking the logarithm of both sides of the equation, we get:
n * log(3) > log(20,000 / a)
Now, let's substitute the value of a in the equation:
n * log(3) > log(20,000 / 5)
Simplifying,
n * log(3) > log(4,000)
Dividing both sides of the equation by log(3):
n > log(4,000) / log(3)
Using a calculator, the value of log(4,000) / log(3) is approximately 6.0731.
Since n represents the number of terms in the sequence, and it must be a whole number, the smallest value of n that satisfies the inequality is 7.
Therefore, to find the first term (a) that exceeds 20,000, we can calculate:
a = 5 * 3^7
Using a calculator, we get:
a = 5 * 2,187
a = 10,935
So, the first term of the sequence that exceeds 20,000 is 10,935.
To find the first term of the sequence 5, 15, 45, 135 that exceeds 20,000, we can start by determining the pattern or rule followed by the sequence.
Looking at the given sequence, we can observe that each term is obtained by multiplying the previous term by 3. This makes it a geometric sequence with a common ratio of 3.
Using this information, we can write the general formula for the nth term of a geometric sequence as:
an = a1 * r^(n-1)
Where:
an = the nth term of the sequence
a1 = the first term of the sequence
r = the common ratio of the sequence
n = the position of the term in the sequence
Now, we can set up the equation to find the first term that exceeds 20,000:
20,000 = a1 * 3^(n-1)
To find the first term that exceeds 20,000, we need to solve this equation by trial and error.
Let's start with assuming a1 = 5 and incrementing n until we find a term that exceeds 20,000:
For n = 1: 20,000 = 5 * 3^(1-1) = 5
For n = 2: 20,000 = 5 * 3^(2-1) = 15
For n = 3: 20,000 = 5 * 3^(3-1) = 45
For n = 4: 20,000 = 5 * 3^(4-1) = 135
For n = 5: 20,000 = 5 * 3^(5-1) = 405
For n = 6: 20,000 = 5 * 3^(6-1) = 1215
For n = 7: 20,000 = 5 * 3^(7-1) = 3645
For n = 8: 20,000 = 5 * 3^(8-1) = 10935
For n = 9: 20,000 = 5 * 3^(9-1) = 32805
Based on the calculations, we see that for n = 9, the term is 32,805, which exceeds 20,000. Therefore, the first term of the sequence that exceeds 20,000 is 5.