I have two left I need help with and don't understand.
Find the partial sum S6
1. An=3n+2
2. Find the number of terms n in an arithmetic sequence if a1=20, d=5, and An=75.
1. To find the partial sum of the sequence given by An = 3n + 2, we need to find S6, which represents the sum of the first 6 terms of the sequence.
To find the value of each term, we substitute the values of n from 1 to 6 into the formula An = 3n + 2:
A1 = 3(1) + 2 = 5
A2 = 3(2) + 2 = 8
A3 = 3(3) + 2 = 11
A4 = 3(4) + 2 = 14
A5 = 3(5) + 2 = 17
A6 = 3(6) + 2 = 20
Now we can find the partial sum S6 by adding up these 6 terms:
S6 = A1 + A2 + A3 + A4 + A5 + A6
= 5 + 8 + 11 + 14 + 17 + 20
= 75
Therefore, the partial sum S6 for the given sequence is 75.
2. To find the number of terms n in an arithmetic sequence given a1 = 20, d = 5, and An = 75, we can use the formula for the n-th term of an arithmetic sequence:
An = a1 + (n - 1) * d
Substituting the given values, we have:
75 = 20 + (n - 1) * 5
Next, we simplify and solve for n:
75 - 20 = (n - 1) * 5
55 = 5(n - 1)
Dividing both sides by 5, we get:
11 = n - 1
Finally, adding 1 to both sides, we find:
n = 12
Therefore, the number of terms in the arithmetic sequence is n = 12.