1. let an=4n find the sum of the first 5 terms.
How would you do this?
2. a1=3; an+1=3an+18. Find a6
How would you do this?
3. You start you 30 year career at a salary of $50,000 and receive yearly raises of $1,000. How much money will you have earned, in total, over the 30 years?
these are all just arithmetic sequences.
An = a + (n-1)d
Sn = n/2 (2a + (n-1)d)
see what this gives you
1. To find the sum of the first 5 terms of the sequence an = 4n, you can use the formula for the sum of an arithmetic series, which is Sn = (n/2)(a1 + an), where n is the number of terms, a1 is the first term, and an is the last term.
In this case, the first term a1 = 4(1) = 4, and the fifth term a5 = 4(5) = 20.
Substituting these values into the formula, we get:
S5 = (5/2)(4 + 20) = (5/2)(24) = 60
Therefore, the sum of the first 5 terms of the sequence an = 4n is 60.
2. To find the value of a6 in the sequence defined by a1 = 3 and an+1 = 3an + 18, we can use the recursive formula to find the subsequent terms of the sequence.
Starting with a1 = 3, we can calculate the values of a2, a3, a4, a5, and finally a6:
a2 = 3a1 + 18 = 3(3) + 18 = 9 + 18 = 27
a3 = 3a2 + 18 = 3(27) + 18 = 81 + 18 = 99
a4 = 3a3 + 18 = 3(99) + 18 = 297 + 18 = 315
a5 = 3a4 + 18 = 3(315) + 18 = 945 + 18 = 963
a6 = 3a5 + 18 = 3(963) + 18 = 2889 + 18 = 2907
Therefore, a6 in the given sequence is 2907.
3. To calculate the total amount of money earned over 30 years with yearly raises, you can use the formula for the sum of an arithmetic series, Sn = (n/2)(a1 + an), where n is the number of terms, a1 is the initial salary, and an is the salary after n years.
In this case, the initial salary a1 is $50,000, and the number of years n is 30. Since the salary increases by $1,000 per year, the salary after n years an can be calculated as a1 + $1,000 * (n - 1).
Substituting these values into the formula, we get:
Sn = (30/2)(50,000 + 50,000 + $1,000 * (30 - 1)) = 15(50,000 + 50,000 + $1,000 * 29) = 15(100,000 + $29,000) = 15($1,450,000)
Therefore, over the 30-year career, a total of $1,450,000 will be earned.