What is the 11th term of the following geometric sequence? Enter a whole number
7/9,-7/3,7,-21,63
a(n+1) = a(n) * -3
a = 7/9
r = -3
a(n) = a r^(n-1)
so
a(11) = (7/9) (-3)^10
= 7 (3)^8
= 45927
To find the 11th term of a geometric sequence, we need to identify the common ratio (r) between consecutive terms.
To find the common ratio, we can divide any term by its previous term. Let's choose the second and third terms:
(-7/3) / (7/9)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. Thus:
(-7/3) * (9/7)
Simplifying this expression gives us:
-9/3
Which simplifies further to:
-3
So, the common ratio (r) for this geometric sequence is -3.
To find the 11th term, we can use the formula for the nth term of a geometric sequence:
An = A1 * r^(n-1)
where:
An is the nth term,
A1 is the first term,
r is the common ratio, and
n is the position of the term we are looking for (11 in this case).
Using the values we have:
A1 = 7/9
r = -3
n = 11
Let's substitute these values into the formula and evaluate it:
A11 = (7/9) * (-3)^(11-1)
= (7/9) * (-3)^10
= (7/9) * (59049)
= 41343
Therefore, the 11th term of this geometric sequence is 41343.
To find the 11th term of the given geometric sequence, we need to know the first term and the common ratio. In a geometric sequence, each term is found by multiplying the previous term by the common ratio.
In this case, we are given the first five terms of the sequence: 7/9, -7/3, 7, -21, and 63. Let's determine the common ratio by dividing any term by its preceding term.
The common ratio (r) is found by dividing any term by its preceding term:
-7/3 ÷ 7/9 = -7/3 * 9/7 = -63/21 = -3
Therefore, the common ratio (r) is -3.
Next, we need to find the first term, which is denoted as "a" in the sequence. To do this, substitute the given values:
The second term (a2) = -7/3 = a * r
We can rearrange this expression to solve for the first term (a):
a = a2 ÷ r = -7/3 ÷ -3 = 7/9
So, the first term (a) is 7/9, and the common ratio (r) is -3.
Now, we can use the formula to find the nth term of a geometric sequence:
an = a * r^(n-1)
Plugging in the values, we get:
a11 = (7/9) * (-3)^(11-1)
= (7/9) * (-3)^10
To simplify (-3)^10, we need to consider the sign. Since we have an even exponent, the result will always be a positive value:
(-3)^10 = 3^10 = 59049
Now, multiply the first term by the simplified value:
a11 = (7/9) * 59049
= 59049/9
= 6550.333...
Since we are asked to enter a whole number, we should round the value to the nearest whole number. Therefore, the 11th term of the given geometric sequence is approximately 6550.