Answer :
- The 5th term of the GP is greater than the 1st term by 215, which can be written as $ar^4 = a + 215$.
- Substitute the common ratio $r = 3$ into the equation: $a(3)^4 = a + 215$.
- Simplify the equation to $81a = a + 215$, and solve for 'a': $80a = 215$.
- The first term is $a = \frac{215}{80} = \frac{43}{16}$, so the final answer is $\boxed{\frac{43}{16}}$.
### Explanation
1. Understanding the Problem
We are given that the common ratio of a geometric progression (GP) is 3. We are also told that the 5th term is greater than the 1st term by 215. Our goal is to find the 1st term of the GP. Let's denote the first term as 'a'.
2. Setting up the Equation
The $n^{th}$ term of a GP is given by $ar^{n-1}$, where a is the first term and r is the common ratio. Therefore, the 5th term is $ar^{4}$. We are given that the 5th term is greater than the 1st term by 215, so we can write the equation $ar^4 = a + 215$.
3. Substituting the Common Ratio
We are given that the common ratio $r = 3$. Substituting this into our equation, we get $a(3)^4 = a + 215$, which simplifies to $81a = a + 215$.
4. Solving for the First Term
Now, we solve for 'a'. Subtracting 'a' from both sides gives $80a = 215$. Dividing both sides by 80, we find $a = \frac{215}{80}$.
5. Simplifying the Fraction
We can simplify the fraction $\frac{215}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives us $a = \frac{43}{16}$. Therefore, the first term of the GP is $\frac{43}{16}$.
6. Final Answer
Thus, the first term of the geometric progression is $\boxed{\frac{43}{16}}$.
### Examples
Geometric progressions are useful in many areas of mathematics and in real-world applications. For example, understanding geometric progressions can help calculate compound interest, where the amount increases by a fixed percentage over time. They are also used in physics to model phenomena such as radioactive decay, where the amount of a substance decreases by a fixed percentage over time. In finance, GPs can help analyze investments that grow at a constant rate.
- Substitute the common ratio $r = 3$ into the equation: $a(3)^4 = a + 215$.
- Simplify the equation to $81a = a + 215$, and solve for 'a': $80a = 215$.
- The first term is $a = \frac{215}{80} = \frac{43}{16}$, so the final answer is $\boxed{\frac{43}{16}}$.
### Explanation
1. Understanding the Problem
We are given that the common ratio of a geometric progression (GP) is 3. We are also told that the 5th term is greater than the 1st term by 215. Our goal is to find the 1st term of the GP. Let's denote the first term as 'a'.
2. Setting up the Equation
The $n^{th}$ term of a GP is given by $ar^{n-1}$, where a is the first term and r is the common ratio. Therefore, the 5th term is $ar^{4}$. We are given that the 5th term is greater than the 1st term by 215, so we can write the equation $ar^4 = a + 215$.
3. Substituting the Common Ratio
We are given that the common ratio $r = 3$. Substituting this into our equation, we get $a(3)^4 = a + 215$, which simplifies to $81a = a + 215$.
4. Solving for the First Term
Now, we solve for 'a'. Subtracting 'a' from both sides gives $80a = 215$. Dividing both sides by 80, we find $a = \frac{215}{80}$.
5. Simplifying the Fraction
We can simplify the fraction $\frac{215}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives us $a = \frac{43}{16}$. Therefore, the first term of the GP is $\frac{43}{16}$.
6. Final Answer
Thus, the first term of the geometric progression is $\boxed{\frac{43}{16}}$.
### Examples
Geometric progressions are useful in many areas of mathematics and in real-world applications. For example, understanding geometric progressions can help calculate compound interest, where the amount increases by a fixed percentage over time. They are also used in physics to model phenomena such as radioactive decay, where the amount of a substance decreases by a fixed percentage over time. In finance, GPs can help analyze investments that grow at a constant rate.
