In an arithmetic sequence, the 6th term is 25 and the 11th term is 45, what is the 20th term?
Answer (1)
Answer: {20th=81}Step-by-step explanation:To find the 20th term of the arithmetic sequence, we can start by considering the information given. In an arithmetic sequence, the \(n\)-th term can be expressed as:{a_n = a + (n-1)d}where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.From the information given:1. The 6th term is 25: {a + 5d = 25(1)} 2. The 11th term is 45: {a + 10d = 45(2)} Now, we can solve this system of equations to find \(a\) and \(d\).First, subtract equation (1) from equation (2):(a + 10d) - (a + 5d) = 45 - 25This simplifies to:{5d = 20}Thus, we can solve for \(d\):{d = \frac(20)(5)= 4}Now that we have \(d\), we can substitute it back into equation (1) to find \(a\):{a + 5(4) = 25}{a + 20 = 25}{a = 25 - 20 = 5}Now we have both \(a\) and \(d\):- \(a = 5\)- \(dTo find the 20th term of the arithmetic sequence, we can start by considering the information given. In an arithmetic sequence, the \(n\)-th term can be expressed as:{a_n = a + (n-1)d}where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.From the information given:1. The 6th term is 25: {a + 5d = 25(1)} 2. The 11th term is 45: {a + 10d = 45(2)} Now, we can solve this system of equations to find \(a\) and \(d\).First, subtract equation (1) from equation (2):{(a + 10d) - (a + 5d) = 45 - 25}This simplifies to:{5d = 20}Thus, we can solve for \(d\):{d = \frac{20}{5} = 4}Now that we have \(d\), we can substitute it back into equation (1) to find \(a\):{a + 5(4) = 25}{a + 20 = 25}{a = 25 - 20 = 5}Now we have both \(a\) and \(d\):- \(a = 5\)- \(d = 4\)Now, we can find the 20th term:{a_{20} = a + (20-1)d = 5 + 19(4)}{= 5 + 76}{= 81}So, the 20th term of the arithmetic sequence is 81.Now, we can find the 20th term:a_{20} = a + (20-1)d = 5 + 19(4)= 5 + 76{20th= 81}So, the 20th term of the arithmetic sequence is 81.I hope this helps u took me long
