Question

If the sum of the first ten terms of the series (1 3/5​)²(2 2/5​)²(3 1/5​)²4²(4 4/5​)2…… is 516​m, then m is equal to:

a. 3
b. 5
c. 6
d. 8

Answer 1

To find the value of m, we need to calculate the sum of the first ten terms of the given series. We set up an equation and calculate each term of the series by converting mixed numbers to improper fractions. We then evaluate the sum of the arithmetic sequence and subtract the sum of the first nine terms to find the 10th term.

Step-by-step explanation:

To find the value of m, we need to calculate the sum of the first ten terms of the given series. The series starts with (1 ⅕)², then (2 ⅛)², then (3 ⅕)², and so on. To calculate each term, we need to convert the mixed numbers to improper fractions. The sum of the first ten terms of the series is 516, so we can set up an equation to solve for m.

First, let's calculate the value of each term:

(1 ⅕)² = (8/5)² = (64/25)

(2 ⅛)² = (17/5)² = (289/25)

(3 ⅕)² = (26/5)² = (676/25)

And so on.

Now, let's set up the equation:

(64/25) + (289/25) + (676/25) + ... + (10th term) = 516

Combining like terms, we have:

(64 + 289 + 676 + ... + (10th term))/25 = 516

Now, let's calculate the sum of the arithmetic sequence:

(64 + 289 + 676 + ... + (10th term)) = (516 · 25) = 12900

Now, we can subtract the sum of the first nine terms from the total sum:

(10th term) = 12900 - [(64 + 289 + 676 + ... + (9th term))]

By evaluating the sum of the first nine terms, we get:

(9th term) = 12900 - (516 · 25 - (64 + 289 + 676))

Simplifying further, we have:

(9th term) = 12900 - (12900 - 1029)

(9th term) = 1029

Finally, we can calculate the 10th term:

(10th term) = 1029 + 516 - 442

(10th term) = 1103

Therefore, m is equal to 1103.