find the 25th term of a geometric sequence having a common ratio r=1/4 and term a=3
for a GS
termn = ar^(n-1)
= 3(1/4)^24
evaluate
Why did the geometric sequence go to therapy? Because it had some common ratio issues!
To find the 25th term of the geometric sequence with a common ratio of 1/4 and a first term of 3, we can use the formula:
\(a_n = a_1 \cdot r^{(n-1)}\)
Plugging in the values, we have:
\(a_{25} = 3 \cdot \left(\frac{1}{4}\right)^{(25-1)}\)
Simplifying this, we get:
\(a_{25} = 3 \cdot \left(\frac{1}{4}\right)^{24}\)
Now, I know math can be a bit boring, but bear with me! Let's compute this to find out the 25th term:
\(a_{25} = 3 \cdot (0.25)^{24}\)
Calculating this, we find:
\(a_{25} \approx 0.00000005\)
So, the 25th term of the geometric sequence is approximately 0.00000005. But seriously, that's a pretty tiny number – it's practically hiding!
To find the 25th term of a geometric sequence with a common ratio of 1/4 and a first term of 3, we can use the formula for the nth term of a geometric sequence: aₙ = a * r^(n-1).
In this case, we have a = 3 and r = 1/4. Plugging these values into the formula, we get:
aₙ = 3 * (1/4)^(25-1)
Simplifying this equation gives us:
aₙ = 3 * (1/4)^24
To evaluate this expression, we can calculate the value of 1/4 raised to the power of 24:
(1/4)^24 = 1 / (4^24)
Using a calculator, we find that 4 raised to the power of 24 is 1,099,511,627,776:
4^24 = 1,099,511,627,776
Therefore, we can substitute this value back into the equation:
aₙ = 3 / 1,099,511,627,776
Calculating this, we get:
aₙ ≈ 2.727 x 10^(-12)
Thus, the 25th term of the geometric sequence is approximately 2.727 x 10^(-12).
To find the 25th term of a geometric sequence, we can use the formula:
An = a * r^(n-1)
where An is the nth term, a is the first term, r is the common ratio, and n is the term number we are trying to find.
In this case, the first term (a) is given as 3 and the common ratio (r) is given as 1/4. We need to find the 25th term (An).
Plugging in these values into the formula, we have:
A25 = 3 * (1/4)^(25-1)
Simplifying the exponent, we have:
A25 = 3 * (1/4)^24
Now, let's evaluate this expression using a calculator or by simplifying the fraction:
A25 = 3 * (1/2^2)^24
A25 = 3 * (1/2^48)
A25 = 3 * (1/281474976710656)
Multiplying the numerator and denominator, we get:
A25 = 3/281474976710656
Therefore, the 25th term of the geometric sequence is 3/281474976710656.