1)find the sum of the odd integers from 25 to 75
2)find the sum of the first 20 terms in the series 10+5+2.5+1.25... .
please show the steps to solve the problem^_^
1) To find the sum of the odd integers from 25 to 75, you can use the formula for the sum of an arithmetic series. The formula is given by:
Sn = (n/2) * (a1 + an),
where Sn is the sum, n is the number of terms, a1 is the first term, and an is the nth term.
In this case, we need to find the sum of the odd integers from 25 to 75, so the first term, a1, is 25, the last term, an, is 75, and the common difference is 2 since we're dealing with odd numbers (25, 27, 29, ...).
To find the number of terms, we can use the formula for finding the nth term in an arithmetic series:
an = a1 + (n - 1) * d,
where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.
In this case, we need to find n such that an = 75:
75 = 25 + (n - 1) * 2.
We can solve this equation for n:
75 - 25 = 2n - 2,
50 = 2n - 2,
52 = 2n,
n = 26.
Now that we know n is 26, we can plug in the values into the sum formula:
Sn = (n/2) * (a1 + an),
= (26/2) * (25 + 75),
= 13 * 100,
= 1300.
Therefore, the sum of the odd integers from 25 to 75 is 1300.
2) To find the sum of the first 20 terms in the series 10 + 5 + 2.5 + 1.25..., we can see that each term is being divided by 2.
To find the sum of a finite geometric series, we can use the formula:
Sn = a1 * (r^n - 1) / (r - 1),
where Sn is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term, a1, is 10, and the common ratio, r, is 1/2 since each term is divided by 2.
Now we need to find n, the number of terms. Since this is the sum of the first 20 terms, n is simply 20.
Plugging in the values into the sum formula:
Sn = a1 * (r^n - 1) / (r - 1),
= 10 * ((1/2)^20 - 1) / (1/2 - 1).
To simplify, we can calculate (1/2)^20 by repeatedly dividing 1 by 2 a total of 20 times. This will give us a very small decimal number.
After calculating (1/2)^20:
Sn = 10 * (0.00000095 - 1) / (-0.5),
= 10 * (-0.99999905) / (-0.5).
Finally, we can simplify the expression:
Sn = 10 * (-0.99999905) / (-0.5),
= 19.999981.
Therefore, the sum of the first 20 terms in the series 10 + 5 + 2.5 + 1.25... is approximately 19.999981.