Moshe would like to get tickets for President inauguration. It will take him 20 years. He decides to pay $200 the first year and the increase his payment by $60 each year.
a) How much did he pay in the last year?
b) How much did he pay altogether for the tickets?
What kind of nonsense is this? Tickets to presidential inaugurations are free.
They are distributed to members of Congress who distribute them to their constituents.
This is the problem for arithmetic sequences and series:
Moshe would like to get tickets for President Justin Bieber's inauguration. It will take him 20 years. He decides to pay $200 the first year and then increase his payment by $60 each year.
a) How much did he pay in the last year?
b) How much did he pay altogether for the tickets?
First year 12*200=2400
61680 total
To find the answers to both questions, we can use the arithmetic series formula. The formula for the nth term of an arithmetic series is given by:
\[ a_n = a_1 + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term of the series,
- \( a_1 \) is the first term of the series,
- \( n \) is the number of terms in the series,
- \( d \) is the common difference between consecutive terms of the series.
In this case, the first term (\( a_1 \)) is $200, and the common difference (\( d \)) is $60. Let's calculate the answers:
a) The last year for Moshe will be the 20th year, so we need to find the 20th term (\( a_{20} \)).
Substituting the values into the formula:
\[ a_{20} = 200 + (20 - 1) \cdot 60 \]
\[ a_{20} = 200 + 19 \cdot 60 \]
\[ a_{20} = 200 + 1140 \]
\[ a_{20} = 1340 \]
So, in the last year, Moshe paid $1340.
b) To find the total amount paid altogether, we can use the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]
Substituting the values into the formula:
\[ S_{20} = \frac{20}{2} \cdot (200 + 1340) \]
\[ S_{20} = 10 \cdot 1540 \]
\[ S_{20} = 15400 \]
Therefore, Moshe paid a total of $15400 altogether for the tickets.