The seventh term of the GP 16/9,-8/3,4,...is _ (A)-9/2 (B)81/4 (C)-22/3 (D)4/3
a = 16/9
r = (-8/3) ÷(16/9)
= (-8/3)(9/16) = -3/2
term (7) = a r^6
= (16/9) (-3/2)^6
= (16/9)(729/64) = ...
To find the seventh term of a geometric progression (GP), we need to use the formula for the nth term of a GP, given by:
an = a1 * r^(n-1),
where:
- an represents the nth term of the GP,
- a1 is the first term of the GP,
- r is the common ratio of the GP,
- n is the term number of interest.
In this case, we have the following information:
- The first term, a1, is 16/9,
- The common ratio, r, can be found by dividing any term by its preceding term: r = (-8/3) / (16/9) = -8/3 * 9/16 = -3/2.
Now, we can use the formula to find the seventh term of the GP:
a7 = (16/9) * (-3/2)^(7-1).
Evaluating this expression:
a7 = (16/9) * (-3/2)^6
= (16/9) * (9/4)
= 36/4
= 9.
Therefore, the seventh term of the GP is 9.
None of the given options in the question matches the correct answer.
Well, let's see if I can come up with a humorous way to solve this!
To find the seventh term of a geometric progression, we need to apply a formula. But math formulas can be scarier than a clown hiding in your closet!
So, buckle up and get ready for some mathematical fun!
The common ratio (r) of this geometric progression can be found by dividing any term by its previous term.
If you divide -8/3 by 16/9, you'll get -1/2. And if you divide 4 by -8/3, you'll get -3/2.
Now, take a deep breath and let's calculate!
To find the seventh term, we start with the third term, which is 4. To get the seventh term, we multiply the third term by the common ratio (r) four times.
4 * (-1/2) * (-1/2) * (-1/2) * (-1/2) = 4/16 * 1/16 = 1/64.
So, the seventh term of the geometric progression is 1/64.
But wait! Don't pick that answer just yet! This is a multiple-choice question, and we need to find the answer among the given options!
So, let's compare our calculated value of 1/64 with the answer choices:
(A) -9/2: Yikes! That's not even close!
(B) 81/4: Nope, not even in the same ballpark!
(C) -22/3: This answer seems a little too negative for our progression!
(D) 4/3: Nope, not even close either!
Well, it looks like none of the options match our calculated value. So, maybe the answer choices have a case of "multiple-choice confusion"!
But fear not! We have solved the mystery. The correct answer is not among the options provided. So, we can safely say that none of the given answers are correct.
Remember, math can sometimes be a clown with a terrible sense of humor! But at least we had some fun along the way!
To find the seventh term of the geometric progression (GP) with the given terms 16/9, -8/3, 4, we need to determine the common ratio (r) first.
The common ratio (r) can be found by dividing any term by its previous term. Let's calculate it:
r = (-8/3) / (16/9)
r = -8/3 * 9/16
r = -72/48
r = -3/2
Now that we have the common ratio (r = -3/2), we can find the seventh term using the formula for the nth term of a GP:
Tn = ar^(n-1)
Here, a is the first term, r is the common ratio, and n is the term we want to find. Let's substitute the values:
T7 = (16/9) * (-3/2)^(7-1)
T7 = (16/9) * (-3/2)^6
T7 = (16/9) * (9/4)
T7 = 144/36
T7 = 4
Therefore, the seventh term of the given geometric progression is 4. The answer is (D) 4/3.