Given the terms A10 = 3/5¹² and A15 = 3/16384 of a geometric sequence, find the exact value of the term A30 of the sequence
A30 = 3/1073741824
To find the exact value of the term A30 of the geometric sequence, you need to determine the common ratio (r) first.
We can find the common ratio (r) by dividing the term A15 by the term A10:
r = A15 / A10
Let's substitute the given values:
r = (3/16384) / (3/5¹²)
To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction:
r = (3/16384) * (5¹²/3)
Next, cancel out common factors:
r = (5¹² / 16384)
Simplify the exponent:
r = (5¹² / (2^14))
Now, let's find the value of the 30th term (A30) using the formula for a geometric sequence:
A30 = A1 * r^(30-1)
But we don't have the value of A1.
However, we can use the formula for the nth term of a geometric sequence to find A1:
A1 = A10 / (r^9)
Let's substitute the known values:
A1 = (3/5¹²) / (5¹² / (2^14))^9
Now, let's simplify:
A1 = (3/5¹²) / (5¹²)^9 * (2^14)^9
A1 = (3/5¹²) / 5^108 * 2^126
A1 = (3/5¹²) / (5^108 * 2^126)
Now that we have the value of A1, we can calculate the 30th term (A30):
A30 = A1 * r^(30-1)
A30 = ((3/5¹²) / (5^108 * 2^126)) * (5¹² / 2^14)^(30-1)
Simplify the exponents:
A30 = ((3/5¹²) / (5^108 * 2^126)) * (5^360 / 2^14)
Now, cancel out common factors:
A30 = 3 / (2^126 * 5^108 * 2^14)
A30 = 3 / (2^(126+14) * 5^108)
Finally, simplify:
A30 = 3 / (2^140 * 5^108)
Thus, the exact value of the term A30 of the geometric sequence is 3 / (2^140 * 5^108).
To find the exact value of the term A30 in the geometric sequence, we need to first determine the common ratio (r) of the sequence.
The common ratio (r) can be calculated by dividing any term by its preceding term. Let's calculate it using the given terms A10 and A15:
r = A15 / A10
r = (3/16384) / (3/5¹²)
r = (3/16384) * (5¹²/3)
r = 5¹² / 16384
Now that we have the common ratio (r), we can use it to find the value of the term A30. We know that:
A30 = A15 * r¹⁵
Plugging in the values:
A30 = (3/16384) * (5¹²/3)¹⁵
Calculating this expression will give us the exact value of the term A30 in the sequence.