Answered

Evaluate the sum:

[tex]\[ 7 \cdot \frac{1}{3 \cdot 5 \cdot 7} + \frac{1}{5 \cdot 7 \cdot 9} + \frac{1}{7 \cdot 9 \cdot 11} + \frac{1}{9 \cdot 11 \cdot 13} + \ldots + \frac{1}{23 \cdot 25 \cdot 27} \][/tex]



Answer :

GinnyAnswer
To solve the given series:

[tex]\[7 \frac{1}{3 \cdot 5 \cdot 7} + \frac{1}{5 \cdot 7 \cdot 9} + \frac{1}{7 \cdot 9 \cdot 11} + \frac{1}{9 \cdot 11 \cdot 13} + \ldots + \frac{1}{23 \cdot 25 \cdot 27},\][/tex]

we will approach it term by term.

1. First Term:
[tex]\[7 \left(\frac{1}{3 \cdot 5 \cdot 7}\right) = 7 \cdot \frac{1}{105}.\][/tex]


2. Subsequent Terms:
For each subsequent term, we compute the fractions:

- Second Term:
[tex]\[\frac{1}{5 \cdot 7 \cdot 9} = \frac{1}{315}.\][/tex]

- Third Term:
[tex]\[\frac{1}{7 \cdot 9 \cdot 11} = \frac{1}{693}.\][/tex]

- Fourth Term:
[tex]\[\frac{1}{9 \cdot 11 \cdot 13} = \frac{1}{1287}.\][/tex]

- Continuing in this manner, each term in the series contributes to the overall summation.

3. Summing the Terms:
The full expression is:
[tex]\[ 7 \cdot \frac{1}{105} + \frac{1}{315} + \frac{1}{693} + \frac{1}{1287} + \ldots + \frac{1}{23 \cdot 25 \cdot 27}. \][/tex]

Adding these fractions together gives us the final sum:

[tex]\[ = 0.08296296296296297 \][/tex]

Therefore, the result for the summation series is:

[tex]\[ \boxed{0.08296296296296297} \][/tex]

Other Questions