Find the sum of the first 10 positive integers

Is it 110?

Which term of the arithmetic sequence is 25?
-2,1,4,...
Is it 70?

Which term of the arithmetic sequence is177?
2,9,16,...
Is it 1234?

Find the sum of the first 8 positive odd integers.
Is it 64?

#1, wrong

1+2+..+8+9+10
= (10/2)(first + last) = 55

#2. wrong
a=-2 , d = 3, term(n) = 25
term(n) = a + (n-1)d
25 = -2 + 3(n-1)
27 = 3n - 3
30 = 3n
n = 10 , so it is the 10th term

#3. wrong
a=2, d = 7, term(n) = 177
177 = 2 + 7(n-1)
175 = 7n - 7
182 = 7n
n = 26 , so the 26th term

#4. yeah, you got one right.
1+3+5+... for 8 terms
a=1, d=2, n=8
sum(8) = (8/2)(2 + 7(2)) = 64
or
notice:
sum(1) = 1 = 1^2
sum(2) = 4 = 2^2
sum(3) = 9 = 3^2
sum(4) = 16 = 4^2
so sum(8) = 8^2 = 64

To find the sum of the first 10 positive integers, you can use the formula for the sum of an arithmetic series: Sn = (n/2)(a + l), where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term. In this case, the first term is 1 and the last term is 10.

Plugging the values into the formula, Sn = (10/2)(1 + 10) = (5)(11) = 55.

Therefore, the sum of the first 10 positive integers is 55, not 110.

To find the term in an arithmetic sequence, you can use the formula for the nth term of an arithmetic sequence: An = a + (n-1)d, where An is the nth term, a is the first term, n is the term number, and d is the common difference between terms.

For the sequence -2, 1, 4,..., the first term is -2 and the common difference is 3. To find which term is 25, we can plug the values into the formula and solve for n:

25 = -2 + (n-1)(3)
Simplifying the equation,
25 = -2 + 3n - 3
25 = 3n - 5
3n = 30
n = 10

Therefore, the 25th term in the sequence is 10, not 70.

For the sequence 2, 9, 16,..., the first term is 2 and the common difference is 7. To find which term is 177, we can plug the values into the formula and solve for n:

177 = 2 + (n-1)(7)
Simplifying the equation,
177 = 2 + 7n - 7
177 = 7n - 5
7n = 182
n = 26

Therefore, the 177th term in the sequence is 26, not 1234.

To find the sum of the first 8 positive odd integers, you can use the formula for the sum of an arithmetic series with a common difference of 2 since the odd integers have a common difference of 2: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.

In this case, the first term is 1, the common difference is 2, and the number of terms is 8.

Plugging the values into the formula, Sn = (8/2)(2(1) + (8-1)(2)) = (4)(2 + 14) = (4)(16) = 64.

Therefore, the sum of the first 8 positive odd integers is indeed 64.