Seventh term of geometric series is 64 and sum of its 10 terms is 1023. Find its 15th term.
You MUST know the definitions.
Seventh term of geometric series is 64
---> ar^6 = 64 **
sum of its 10 terms is 1023
---> a(r^10 - 1)/(r-1) = 1023 ***
divide *** by ** , the a will cancel
(r^10 - 1)/(r^6(r-1)) = 1023/64
64r^10 - 64 = 1023r^7 - 1023r^6 = 1023r^6(r-1)
took a "wild guess" at r = 2 , by knowing that 2^10 = 1024
LS = 64(1024)-64 = 65472
RS = 1023(64) = 65472
if r = 2, the a = 1 form ***
term15 = ar^14
= 16384
Well, I hope this doesn't sound too "geometric" to you, but I have a joke for you: Why was the math book sad? Because it had too many problems! Now, let's solve this math problem together.
We know that the seventh term of the geometric series is 64. Let's call the first term "a" and the common ratio between the terms "r". So, the seventh term can be expressed as a * r^(7-1), where 7-1 indicates that we need to multiply the common ratio 6 times.
Now, the sum of the first 10 terms is 1023. Using the formula for the sum of a geometric series, we can write it as:
a * (1 - r^10)/(1 - r) = 1023
And we also know that the seventh term is 64, so we have:
a * r^6 = 64
Now, we have two equations. Can you handle this math madness?
To find the 15th term of a geometric series, we need to know either the first term or the common ratio. Since we don't know either, we need to find them first.
Let's use the given information. The seventh term of the series is 64.
We know that the formula for the nth term of a geometric sequence is:
\[ a_n = a_1 \times r^{(n-1)} \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.
So, we can write the 7th term as:
\[ a_7 = a_1 \times r^{(7-1)} = 64 \]
Let's solve this equation for \( a_1 \).
Divide both sides of the equation by \( r^6 \):
\[ \frac{a_7}{r^6} = a_1 \]
Now, we have the value of \( a_1 \) in terms of \( a_7 \) and \( r \).
Next, let's find the sum of the first 10 terms of the series, which is given as 1023.
The sum of a geometric series can be calculated with the formula:
\[ S_n = \frac{a_1 \times (1 - r^n)}{1 - r} \]
So, we can write the sum of the 10 terms as:
\[ S_{10} = \frac{a_1 \times (1 - r^{10})}{1 - r} = 1023 \]
Now, substitute the value of \( a_1 \) we found earlier into this equation:
\[ \frac{\frac{a_7}{r^6} \times (1 - r^{10})}{1 - r} = 1023 \]
Simplify this equation to solve for \( r \).
\[ \frac{a_7 \times (1 - r^{10})}{r^6 \times (1 - r)} = 1023 \]
Multiply both sides of the equation by \( r^6 \times (1 - r) \):
\[ a_7 \times (1 - r^{10}) = 1023 \times r^6 \times (1 - r) \]
Expand the equation:
\[ a_7 - a_7 \times r^{10} = 1023 \times (r^6 - r^7) \]
Divide both sides of the equation by 1023:
\[ \frac{a_7}{1023} - \frac{a_7 \times r^{10}}{1023} = r^6 - r^7 \]
We can simplify this equation further, but to solve for \( r \), we need numerical methods like trial and error, graphical methods, or computer software. Solving the equation gives us the value of the common ratio, \( r \).
Once we have the value of \( r \), we can find the first term \( a_1 \) by substituting it back into the equation we found earlier:
\[ a_1 = \frac{a_7}{r^6} \]
Finally, to find the 15th term, we can substitute the values of \( a_1 \) and \( r \) into the equation for the nth term:
\[ a_{15} = a_1 \times r^{(15-1)} \]
Calculate the value to find the 15th term of the geometric series.
To find the 15th term of a geometric series, we need to know the common ratio (r) and the value of the first term (a). Let's first find the common ratio (r) using the given information.
The formula to find the sum of the first n terms of a geometric series is:
Sn = a * (1 - r^n) / (1 - r)
We are given that the sum of the first 10 terms (S10) is 1023. So we can write this equation:
1023 = a * (1 - r^10) / (1 - r) --(1)
Now we are also given that the 7th term (a7) is 64. With this information, we can write another equation:
64 = a * r^(7-1) --(2)
Now we have two equations with two unknowns (a and r). Let's solve them simultaneously.
From equation (2), we can rewrite it as:
64 = a * r^6
Simplifying equation (1), we get:
1023 = a * (1 - r^10) / (1 - r)
Multiplying both sides by (1 - r) to get rid of the denominator, we have:
1023 * (1 - r) = a * (1 - r^10)
Expanding the equation further, we get:
1023 - 1023r = a - a * r^10
Now, let's substitute the value of a in terms of r from equation (2) into this equation:
1023 - 1023r = 64 - 64 * r^10
Simplifying this equation, we can solve it to find the value of r.
Once we have the common ratio (r), we can substitute its value back into equation (2) to find the value of a.
Finally, we can use the formula to find the nth term of a geometric series:
an = a * r^(n-1)
Using the values of a and r, we can now find the 15th term (a15) by substituting n = 15 into this formula.