The sixth term of GP is 16 and the third term is 2.Find the first term and common ratio. 6th term =16 3rd term=2 So ar^5=16 and ar^2=2, then divide. So r^3=8, and r^3 =2^3. Therefore r =2 and a=1/2
The formula of the GP
Looks good to me.
Thanks 😊
The 3rd term of a geometric profession is 16, while the 6th term of the same go is 2 find the first term
Let's use the formula for the nth term of a geometric sequence:
a₃ = a * r²
a₆ = a * r⁵
We can use these equations to create a system of equations:
16 = a * r²
2 = a * r⁵
We can solve for the first term (a) by isolating it in one of the equations:
a = 16 / r²
Substituting this expression for a into the second equation:
2 = (16 / r²) * r⁵
Simplifying:
2 = 16r³
r³ = 1/8
r = 1/2
Now we can find the first term:
16 = a * (1/2)²
16 = a * 1/4
a = 64
Therefore, the first term of the geometric sequence is 64.
Well, it seems like this geometric progression could use a little humor to brighten it up. Let's see if I can come up with something for you:
Why did the geometric progression go to the doctor?
Because it wasn't feeling proportional! It needed a little R&R (that's ratio and repetition)!
But in all seriousness, let's calculate the first term and common ratio. You've got that the 6th term is 16 and the 3rd term is 2. Great!
But I'm going to do things my own funny way, so let me tell you a little story. Once upon a time, two numbers, let's call them a and r, went on a date. They had some exponential chemistry going on. The third term of their relationship was 2, and let me tell you, they were multiplying fast!
By the time they got to their sixth term, it was a whopper of a number, 16! Wowza! Now, let's dive into some math magic to find out the values of a and r.
If we look at the sixth term, we've got ar^5 = 16. And if we look at the third term, we've got ar^2 = 2. Now, let's do some division, my friend:
(ar^5)/(ar^2) = 16/2
That simplifies to:
r^3 = 8
And guess what? 8 happens to be 2^3! So, we can conclude that r = 2.
Now, let's find the first term. We can use the equation ar^2 = 2 and plug in r = 2:
a(2^2) = 2
Simplify that bad boy:
4a = 2
And finally, divide both sides by 4:
a = 1/2
Ta-da! The first term is 1/2 and the common ratio is 2. I hope that puts a smile on your face, my friend. Keep those geometric progressions laughing!
To find the first term (a) and the common ratio (r) of a geometric progression (GP) when given the 6th term and the 3rd term, follow these steps:
Step 1: Use the formula for the nth term of a GP:
The nth term of a GP can be represented as: a * r^(n-1), where 'a' is the first term and 'r' is the common ratio.
Step 2: Set up equations using the given information:
According to the given information, the 6th term (T6) is 16, and the 3rd term (T3) is 2. So, we can write the following equations:
T6 = a * r^(6-1) = 16 (equation 1)
T3 = a * r^(3-1) = 2 (equation 2)
Step 3: Solve the equations using substitution:
Since we have two equations and two unknowns (a and r), we can solve them simultaneously to find the values.
From equation 1: a * r^5 = 16 (equation 3)
From equation 2: a * r^2 = 2 (equation 4)
Step 4: Divide equation 3 by equation 4:
By dividing equation 3 by equation 4, we get:
(r^5) / (r^2) = 16 / 2
Simplifying:
r^(5-2) = 8
r^3 = 8
Step 5: Solve for r:
Taking the cube root of both sides:
r = ∛8
Simplifying:
r = 2
Step 6: Substitute the value of r into equation 2 to find a:
From equation 2: a * r^2 = 2
Plugging in the value of r (2):
a * (2^2) = 2
Simplifying:
4a = 2
Dividing both sides by 4:
a = 2 / 4
Simplifying:
a = 1/2
Therefore, the first term (a) of the GP is 1/2 and the common ratio (r) is 2.