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🟢Which one of the follwing is the formula for the area of a triangle?
🟢What is the process of combining two atomic nuclei to form a heavier nucleus called?
🟢Which of the following is an example of a simple machine?
🟢Which planet in our solar system is known as the "Red Planet"?
🟢What is the process of converting a liquid into a gas called?
🟢Which of the following is an example of a renewable energy source?
📝Solution to Question8️⃣👆
👉The 5th term of an arithmetic sequence is 17 and the 10th term is 32. What is the sum of the first 20 terms of the sequence?
✅Solution: we can use the formula for the nth term of an arithmetic sequence to find the common difference, d:
a5 = a1 + 4d = 17
a10 = a1 + 9d = 32
Subtracting the first equation from the second, we get:
5d = 15
So, d = 3.
📚Now we can use the formula for the sum of the first n terms of an arithmetic sequence:
Sn = n/2 * (a1 + an)
We know that a5 = 17, so we can find a1:
a5 = a1 + 4d = 17
a1 = 17 - 4d = 5
Now we can find a20:
a20 = a1 + 19d = 5 + 19(3) = 62
Finally, we can find the sum of the first 20 terms:
S20 = 20/2 (a_1 + a_20)
= 10 (5 + 62) = 670
Therefore, the sum of the first 20 terms of the arithmetic sequence is 670.
1️⃣2️⃣What is the value of the sith term of the geometric progression 3, 6, 12, 24, ...?
1️⃣1️⃣. What is the common ratio of the geometric progression 2, 4, 8, 16, ...?
1️⃣0️⃣ Which of the following is a geometric progression?
📚Note on Geometric progression:
✅️Definition, Formulas, Applications:
👉Geometric progression, also known as a geometric sequence is sequence of numbers where each term is found by multiplying the previous term by a constant ratio(r).
Formulas:
🟢General Term: In a geometric progression, the general formula for finding the nth term (an) is given by:
an = a * r^(n-1)
where "a" is the first term of the sequence, "r" is the common ratio, and "n" represents the position of the term.
🟢Sum of Terms: The sum of the first n terms (Sn) in a geometric progression can be calculated using the formula:
Sn = a * (1 - r^n) / (1 - r)
Here, "a" is the first term, "r" is the common ratio, and "n" is the number of terms being summed.
Applications:
✅️Finance and Investments: Geometric progressions are widely used in finance to analyze compound interest, where the amount of money grows exponentially over time. They help in understanding the growth or decay of investments, loans, and other financial calculations.
✅️Population Growth: Geometric progressions can be used to model the growth or decline of populations. For example, if the number of individuals in a population doubles each year, a geometric progression can be used to predict the population size at any given time.
✅️Exponential Decay and Growth: Geometric progressions are closely related to exponential functions, which have widespread applications in various fields like physics, chemistry, biology, and economics. They can be used to describe radioactive decay, drug dosage decay, bacterial growth, and more.
Examples:
1️⃣Consider a geometric progression with a first term (a) of 2 and a common ratio (r) of 3. The first few terms of this sequence would be: 2, 6, 18, 54, ...
2️⃣If we have a geometric progression with a first term of 5 and a common ratio of 0.5, the sequence would be: 5, 2.5, 1.25, 0.625, ...
3️⃣Suppose a bank offers an interest rate of 4% per year. If you deposit $1,000, the amount in your account after 5 years can be calculated using a geometric progression. The nth term would represent the amount in the account after n years.
🟢Details of Example 3️⃣
We have a situation where a bank offers an interest rate of 4% per year, and you deposit $1,000. The amount in your account after 5 years can be calculated using a geometric progression.
First, we need to identify the key elements of a geometric progression:
First term (a): The initial amount you deposit, which is $1,000 in this case.
Common ratio (r): The constant rate of growth or decay. In this case, the interest rate is 4%, which can be expressed as a decimal as 0.04. Therefore, the common ratio is 1 + 0.04 = 1.04 (since the amount in the account increases by 4% each year).
The general formula for finding the nth term in a geometric progression is:
an = a * r^(n-1)
Applying the formula to our example, we can find the amount in the account after 5 years (n = 5):
a5 = 1,000 * 1.04^(5-1)
= 1,000 * 1.04^4
≈ 1,000 * 1.16985856
≈ $1,169.86
So, after 5 years, the amount in your account would be approximately $1,169.86.
✅In this context, the sequence of amounts in the account forms a geometric progression, where each term is obtained by multiplying the previous term by a constant ratio of 1.04.
✅The initial deposit of $1,000 serves as the first term, and the subsequent terms represent the amount in the account after each year.
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9️⃣ The sum of the first n terms of an arithmetic sequence is given by the formula Sn = 5n^2 + 2n. What is the 12th term of the sequence?
