Find the no. of terms

1. 3,5,7...33
2.-5,-1,3,...75

Solve each problem.
1. In the Arithmetic sequence 5,8,11,14.... what is n if a sub n is 32?
2.Find the sum of the first 20 terms of the arithmetic sequence -15,-7,31,43...

1. term(n) = a+(n-1)d

you have a = 3, d = 2 , n = ??
33 = 3 + (n-1)(2)
30 = 2n - 2
28 = 2n
n = 14 ------ there are 14 terms

do the 2nd one the same way

for the other two, I don't know how you defined a sub n

for the last:
first find the number of terms in the same way I showed you
you have a = -15 , d = 12 , n = ?
once you find a and d
use
Sum(n) = (n/2)(2a + (n-1)(d) )

Thank you so much . 1. In the Arithmetic sequence 5,8,11,14.... what is n if an is 32?

1. In the Arithmetic sequence 5,8,11,14.... what is n if n is 32?

To find the number of terms in each given sequence, we need to determine the common difference and use the formula for the nth term of an arithmetic sequence.

1. For the sequence 3, 5, 7,..., 33:
- We can see that the common difference is 2. (Each term is obtained by adding 2 to the previous term.)
- The nth term can be calculated using the formula 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, a1 = 3 and an = 33. We need to find the value of n.
- Plugging in the values into the formula, we get: 33 = 3 + (n-1)2.
- Simplifying the equation: 33 = 3 + 2n - 2.
- Combining like terms, we get: 2n + 1 = 33 - 3.
- Simplifying further: 2n + 1 = 30.
- Subtracting 1 from both sides: 2n = 29.
- Dividing both sides by 2: n = 29/2.
- Therefore, the number of terms is 29/2 or 14.5 terms. However, since the number of terms in a sequence must be a whole number, we round down to the nearest whole number.
- The number of terms in the sequence is 14.

2. For the sequence -5, -1, 3,..., 75:
- We can see that the common difference is 4. (Each term is obtained by adding 4 to the previous term.)
- Using the same formula as before, we can calculate the nth term.
- In this case, a1 = -5 and an = 75. We need to find the value of n.
- Plugging in the values into the formula, we get: 75 = -5 + (n-1)4.
- Simplifying the equation: 75 = -5 + 4n - 4.
- Combining like terms, we get: 4n - 9 = 75 + 5.
- Simplifying further: 4n - 9 = 80.
- Adding 9 to both sides: 4n = 89.
- Dividing both sides by 4: n = 89/4.
- Therefore, the number of terms is 89/4 or 22.25 terms. Again, we round down to the nearest whole number.
- The number of terms in the sequence is 22.

Now, let's solve the two additional problems:

1. In the arithmetic sequence 5, 8, 11, 14, ...., what is n if a sub n is 32?
- Again, we use the formula for the nth term of an arithmetic sequence: an = a1 + (n-1)d.
- In this case, a1 = 5 and an = 32. We need to find the value of n.
- Plugging in the values, we get: 32 = 5 + (n-1)3.
- Simplifying the equation: 32 = 5 + 3n - 3.
- Combining like terms, we get: 32 = 3n + 2.
- Subtracting 2 from both sides: 30 = 3n.
- Dividing both sides by 3: n = 30/3.
- Therefore, the value of n is 10.

2. Find the sum of the first 20 terms of the arithmetic sequence -15, -7, 31, 43, ...
- We can see that the common difference is 8. (Each term is obtained by adding 8 to the previous term.)
- The sum of the first n terms of an arithmetic sequence can be calculated using the formula Sn = (n/2)(a1 + an), where Sn is the sum of the first n terms.
- In this case, a1 = -15, an = 43, and n = 20.
- Plugging in the values into the formula, we get: Sn = (20/2)(-15 + 43).
- Simplifying the equation: Sn = 10(-15 + 43).
- Calculating inside the parentheses: Sn = 10(28).
- Multiplying: Sn = 280.
- Therefore, the sum of the first 20 terms is 280.