1. What is the tenth term of the geometric sequence whose first term (a1) is 16 and the common ratio (r) is –½?
2. Given the terms a10 = 3 / 512 and
a15 = 3 / 16384 of a geometric sequence, find the exact value of the first term of the sequence.
1
term(n) = ar^(n-1)
term(10) = 16(-1/2)^9
= 16(-1/512)
= - 1/32
2.
ar^9 = 3/512
ar^14 = 3/16384
divide the 2nd by the first
r^5 = 1/32
r = 1/2
back into the first
a(1/2)^9 = 3/512
a(1/512) = 3/512
a = 3(512)/512 = 3
To find the tenth term of a geometric sequence, 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.
1. Using the given information, the first term (a1) is 16 and the common ratio (r) is -1/2. We want to find the tenth term (a10), so we can substitute this into the formula:
a10 = 16 * (-1/2)^(10-1)
= 16 * (-1/2)^9
= 16 * (-1/512)
= -16/512
= -1/32
Therefore, the tenth term of the geometric sequence is -1/32.
To find the first term of a geometric sequence, we can use the formula for the nth term of a geometric sequence and solve for 'a1':
an = a1 * r^(n-1)
2. Using the given information, we have the following equations:
a10 = 3 / 512
a15 = 3 / 16384
We need to find 'a1', so we can set up two equations:
a10 = a1 * r^(10-1)
a15 = a1 * r^(15-1)
We want to find the exact value of the first term, so we can divide the two equations:
(a10 / a15) = (a1 * r^(10-1)) / (a1 * r^(15-1))
(3/512) / (3/16384) = r^(9-14)
(3/512) * (16384/3) = r^(-5)
16384 / 512 = 1/r^5
32 = 1/r^5
To find the value of 'r', we can take the reciprocal:
r^5 = 1/32
r = (1/32)^(1/5)
Using a calculator, we find that r ≈ 0.5.
Now that we have the value of 'r', we can substitute it into one of the equations to find 'a1':
a10 = a1 * (0.5)^(10-1)
3/512 = a1 * (0.5)^9
3/512 = a1 * (1/512)
3 = a1
Therefore, the exact value of the first term of the sequence is 3.
To find the tenth term of a geometric sequence, we can use the formula:
an = a1 * r^(n-1)
where "an" represents the nth term, "a1" is the first term, "r" is the common ratio, and "n" is the term number.
1. For the given geometric sequence with a1 = 16 and r = -1/2, we can substitute these values into the formula to find the tenth term:
a10 = 16 * (-1/2)^(10-1)
a10 = 16 * (-1/2)^9
a10 = 16 * (-1/512)
a10 = -8/32
a10 = -1/4
Therefore, the tenth term of the geometric sequence is -1/4.
To find the first term of a geometric sequence, we can use the formula:
a1 = (an) / (r^(n-1))
2. For the given geometric sequence with a10 = 3/512 and a15 = 3/16384, we can use these values to find the first term:
a1 = (a10) / (r^(10-1))
a1 = (3/512) / (r^9)
a1 = (3/512) / ((3/16384)^9)
To simplify this expression, we need to find a common denominator for the powers of 3:
a1 = (3/512) / ((3^9) / (16384^9))
a1 = (3/512) * ((16384^9) / (3^9))
Next, we can simplify the fraction by dividing the numerator and denominator by the power of 3:
a1 = (3/512) * (16384^9 / 3^9)
a1 = (3/512) * (16384^9 / 19683)
a1 = (3 * 16384^9) / (512 * 19683)
This gives us the exact value of the first term of the sequence, which may be a very large number.