Find the 10th term in an AP where the first term is 3 and whose 1st, 4th and 13th terms form a GP.
a = 3
Since the GP has a common ratio
(a+3d)/(a) = (a+12d)/(a+3d)
(3+3d)/3 = (3+12d)/(3+3d)
1+d = (1+4d)/(1+d)
d = 2
a+9d = 3+9*2 = 21
Well, looks like this arithmetic progression (AP) is playing a little game with us by forming a geometric progression (GP) within itself. That's quite clever of it!
Let's try to solve this problem with a little humor, shall we?
To find the 10th term of this AP, we'll need to know the common difference. But before we get into calculations, let's meet the GP formed by the 1st, 4th, and 13th terms.
If the 1st, 4th, and 13th terms form a GP, let's call them GP-1, GP-2, and GP-3 respectively. So, GP-2 is the 4th term, and GP-3 is the 13th term.
Now, let's put on our GP detective hats and calculate the common ratio of GP-2 and GP-3. We can do this by dividing the terms:
Common ratio = GP-3 / GP-2
But wait! If it's a GP, then we can simplify things by assuming that GP-2 is usually called GP-1. Confusing, I know, but bear with me.
So, Common ratio = GP-3 / GP-1
Now, armed with this common ratio, let's put on our AP investigator hats and calculate the common difference.
Common difference = GP-1 - 1st Term
So, let's calculate the common ratio and common difference:
Common ratio = GP-3 / 1st Term
Common difference = GP-1 - 1st Term
Once we have the common difference, we can easily find the 10th term using the formula for an AP:
10th term = 1st Term + (10 - 1) * Common Difference
And voila! We have the 10th term!
Disclaimer: The clown inside me might have taken over for the explanation, so please handle this question with care and ignore any clownish behavior. But hey, at least we had some fun, right?
Let's solve this step by step!
Step 1: Finding the common difference (d) of the Arithmetic Progression (AP).
Given that the first term (a1) of the AP is 3, we can assume it as a1 = 3.
Let's assume that the common difference is d.
Step 2: Finding the terms of the GP.
The 1st, 4th, and 13th terms of the AP form a geometric progression (GP). Let's find those terms.
The 1st term of the GP can be obtained by substituting n = 1 in the AP: a1 + (1 - 1)d = a1.
So, the first term of the GP is the first term of the AP, which is 3.
Similarly, the 4th term of the GP can be obtained by substituting n = 4 in the AP: a1 + (4 - 1)d = a4.
So, the term a4 of the AP is the 4th term of the GP.
The 13th term of the GP can be obtained by substituting n = 13 in the AP: a1 + (13 - 1)d = a13.
So, the term a13 of the AP is the 13th term of the GP.
Step 3: Finding the common ratio (r) of the GP.
To find the common ratio (r), we can use the formula: r = (a4 / a1) = (a13 / a4).
Step 4: Finding the terms of the AP using the given information.
Given that the 1st, 4th, and 13th terms form a GP, we know that the common ratio (r) = (a4 / a1) = (a13 / a4).
So, we can find a4 and a13 using this information.
Step 5: Finding the 10th term of the AP.
Now that we have the common difference (d) and the first term (a1) of the AP, we can find the 10th term.
The 10th term (an) of an AP can be calculated using the formula: an = a1 + (n - 1)d.
Let's summarize the steps:
Step 1: Find the common difference (d) of the AP.
Step 2: Find the terms of the GP using the 1st, 4th, and 13th terms of the AP.
Step 3: Find the common ratio (r) of the GP.
Step 4: Find the terms a4 and a13 of the AP using the common ratio (r).
Step 5: Find the 10th term (a10) of the AP using the formula: an = a1 + (n - 1)d.
Now, let's calculate step by step:
To find the 10th term of an arithmetic progression (AP), we need to determine the common difference (d) between consecutive terms.
Let's first find the common difference by using the given information. We are told that the 1st, 4th, and 13th terms form a geometric progression (GP).
Let's label the 1st term as a, the common ratio of the GP as r, and the common difference of the AP as d.
The 1st term of the AP is given as 3, so we have a = 3.
Using the terms mentioned, we can set up the following equations:
a * r^3 = a + 3d (1st term of GP equation)
a * r^12 = a + 12d (4th term of GP equation)
Substituting a = 3 into both equations:
3 * r^3 = 3 + 3d ...(Equation 1)
3 * r^12 = 3 + 12d ...(Equation 2)
Now, let's solve these two equations simultaneously to find the values of r and d.
Dividing Equation 2 by Equation 1, we get:
(r^12) / (r^3) = (3 + 12d) / (3 + 3d)
r^9 = (3 + 12d) / (3 + 3d) ...(Equation 3)
To eliminate d from Equation 3, let's multiply both sides by (3 + 3d):
(r^9) * (3 + 3d) = 3 + 12d
3r^9 + 3d * r^9 = 3 + 12d
3r^9 - 3 = 12d - 3d
3r^9 - 3 = 9d
Now, let's express d in terms of r:
d = (3r^9 - 3) / 9
Therefore, the common difference (d) is equal to (3r^9 - 3) / 9.
Now that we have found the common difference, we can proceed to find the 10th term of the AP.
The formula to find the nth term of an AP is given by:
an = a + (n - 1) * d
Substituting the given values:
a = 3 (1st term)
n = 10
We already know d in terms of r:
d = (3r^9 - 3) / 9
Plugging in the values:
a10 = 3 + (10 - 1) * ((3r^9 - 3) / 9)
= 3 + 9 * (3r^9 - 3) / 9
= 3 + (3r^9 - 3)
= 3r^9
Therefore, the 10th term of the arithmetic progression is 3r^9.