Q14. - Algebra 2

Question

The African-American festival of Kwanzaa includes a ritual involving candles. The first night, a candle is lit and then blown out. The second night, a new candle and the candle from the previous night are lit and blown out. This pattern of lighting a new candle and relighting all the candles from the previous nights is continued for seven nights. Use a formula from this lesson to find the total number of candle lightings during the festival.

Step-by-Step Solution

Answer

The total number of candle lightings is 28.

Step 1. Given Information.

Given, the African-American festival of Kwanzaa includes a ritual involving candles. The first night, a candle is lit and then blown out. The second night, a new candle and the candle from the previous night are lit and blown out. This pattern of lighting a new candle and relighting all the candles from the previous nights is continued for seven nights.

Step 2. Calculation .

The arithmetic series for the lightings is given by 1+2+…

 

The sum  $S_n$ of the first n terms of an arithmetic series is given by   

 $S_n=\frac{n}{2}(2a_1+(n-1)d)$

Here, $a_1=1,d=1,n=7$ 

Plugging the values:

$\begin{array}{l}{S}_n=\frac{n}{2}(2a_1+(n-1)d)\\ S_7=\frac{7}{2}(2\left(1\right)+(7-1)\left(1\right))\\ S_7=\frac{7}{2}(2+6)\\ S_7=\frac{7}{2}\left(8\right)\\ S_7=28\end{array}$

Step 3. Conclusion .

Hence, the total number of candle lightings is 28.