0000:00:00:00:00Start19Number 19ExplanationGiven ana_nan is the n-th term of a sequence of numbers. If an=an+1+3an+3a_n = a_{n+1} + 3a_{n+3}an=an+1+3an+3, a1=−1a_1 = -1a1=−1, and a2=2a_2 = 2a2=2, then the value of a4+a1a_4 + a_1a4+a1 is−2-2−2−1-1−1000111222ExplanationWe are given a recursive formula for a sequence of numbers: an=an+1+3an+3a_n = a_{n+1} + 3a_{n+3}an=an+1+3an+3 We also know that a1=−1a_1 = -1a1=−1 and a2=2a_2 = 2a2=2. We are asked to find the value of a4+a1a_4 + a_1a4+a1. Determining the Value of the Fourth Term To find a4a_4a4, we can use the given equation by substituting n=1n = 1n=1: a1=a1+1+3a1+3a1=a2+3a4\begin{aligned} a_1 &= a_{1+1} + 3a_{1+3} \\ a_1 &= a_2 + 3a_4 \end{aligned}a1a1=a1+1+3a1+3=a2+3a4 Next, we substitute the values a1=−1a_1 = -1a1=−1 and a2=2a_2 = 2a2=2 into the equation: −1=2+3a43a4=−1−23a4=−3a4=−33a4=−1\begin{aligned} -1 &= 2 + 3a_4 \\ 3a_4 &= -1 - 2 \\ 3a_4 &= -3 \\ a_4 &= \frac{-3}{3} \\ a_4 &= -1 \end{aligned}−13a43a4a4a4=2+3a4=−1−2=−3=3−3=−1 Calculating the Sum Now that we have found a4=−1a_4 = -1a4=−1, we can calculate a4+a1a_4 + a_1a4+a1: a4+a1=(−1)+(−1)=−2\begin{aligned} a_4 + a_1 &= (-1) + (-1) \\ &= -2 \end{aligned}a4+a1=(−1)+(−1)=−2 So, the value of a4+a1a_4 + a_1a4+a1 is −2-2−2.
19Number 19ExplanationGiven ana_nan is the n-th term of a sequence of numbers. If an=an+1+3an+3a_n = a_{n+1} + 3a_{n+3}an=an+1+3an+3, a1=−1a_1 = -1a1=−1, and a2=2a_2 = 2a2=2, then the value of a4+a1a_4 + a_1a4+a1 is−2-2−2−1-1−1000111222ExplanationWe are given a recursive formula for a sequence of numbers: an=an+1+3an+3a_n = a_{n+1} + 3a_{n+3}an=an+1+3an+3 We also know that a1=−1a_1 = -1a1=−1 and a2=2a_2 = 2a2=2. We are asked to find the value of a4+a1a_4 + a_1a4+a1. Determining the Value of the Fourth Term To find a4a_4a4, we can use the given equation by substituting n=1n = 1n=1: a1=a1+1+3a1+3a1=a2+3a4\begin{aligned} a_1 &= a_{1+1} + 3a_{1+3} \\ a_1 &= a_2 + 3a_4 \end{aligned}a1a1=a1+1+3a1+3=a2+3a4 Next, we substitute the values a1=−1a_1 = -1a1=−1 and a2=2a_2 = 2a2=2 into the equation: −1=2+3a43a4=−1−23a4=−3a4=−33a4=−1\begin{aligned} -1 &= 2 + 3a_4 \\ 3a_4 &= -1 - 2 \\ 3a_4 &= -3 \\ a_4 &= \frac{-3}{3} \\ a_4 &= -1 \end{aligned}−13a43a4a4a4=2+3a4=−1−2=−3=3−3=−1 Calculating the Sum Now that we have found a4=−1a_4 = -1a4=−1, we can calculate a4+a1a_4 + a_1a4+a1: a4+a1=(−1)+(−1)=−2\begin{aligned} a_4 + a_1 &= (-1) + (-1) \\ &= -2 \end{aligned}a4+a1=(−1)+(−1)=−2 So, the value of a4+a1a_4 + a_1a4+a1 is −2-2−2.
