1 + 1/4 + (1.3)/(4.8)
+ (1.3.5.) + .... = _________
To find the value of the given expression, let's break it down step by step:
1. The first term in the expression is 1.
2. The second term is 1/4, which is equivalent to 0.25.
3. The third term is (1.3)/(4.8). To evaluate this, divide 1.3 by 4.8: 1.3/4.8 = 0.2708 (rounded to 4 decimal places).
4. The fourth term is (1.3)(1.5). Multiply 1.3 and 1.5 to get 1.95.
5. Continue the pattern for subsequent terms: multiplying the previous term by the next number in the series.
To calculate the sum of these terms, add them all together:
1 + 1/4 + (1.3)/(4.8) + (1.3)(1.5) + ...
= 1 + 0.25 + 0.2708 + 1.95 + ...
The series continues indefinitely, so it is important to determine if the series converges or diverges. In this case, it appears to be an infinite geometric series, which can be expressed in the general form:
a + ar + ar^2 + ar^3 + ...
where "a" is the first term and "r" is the common ratio between consecutive terms.
In our case, the common ratio is (1.3)(1.5). Let's call it "r."
If the magnitude of "r" is less than 1, then the series converges and has a finite sum. If the magnitude of "r" is equal to or greater than 1, then the series diverges and does not have a finite sum.
To determine if the series converges or diverges:
1. Calculate the magnitude of "r" by multiplying the first two terms: (1.3)(1.5) = 1.95.
Since the magnitude of "r" is greater than 1, we can conclude that the series diverges and does not have a finite sum.
Hence, we cannot determine the numerical value of the given expression since the series does not converge.