Question

Find the first term given two terms from an arithmetic sequence.a_6 = 12 and a_{14} = 28the common difference is Answerand a_1= Answer

Answer 1

Answer:

the common difference is 2

and a₁ = 2

We need to find the first term given two terms of an arithmetic sequence:

a₆ = 12

a₁₄ = 28

First, we will solve for the common difference 'd'. To do this, we will create a system of equations using the given terms and the following formula:

$a_n=a_1+\lparen n-1)d$

From the given, we can have:

$\begin{gathered} a_6=a_1+\operatorname{\lparen}6-1)d \\ 12=a_1+5d-------Eq1 \\ \end{gathered}$$\begin{gathered} a_{14}=a_1+\operatorname{\lparen}14-1)d \\ 28=a_1+13d \\ a_1=28-13d------Eq2 \end{gathered}$

We will then substitute Eq.2 with Eq.1 to solve for the common difference 'd'

$\begin{gathered} 12=a_1+5d----Eq.1 \\ a_1=28-13d----Eq.2 \\ 12=\left(28-13d\right)+5d \\ 12=28-13d+5d \\ 12-28=-13d+5d \\ -16=-8d \\ d=2 \end{gathered}$

We now have a common difference d = 2

We can now solve the first term of the arithmetic sequence using any of the equations that we had. In this case, let us use Eq.2

$\begin{gathered} \begin{equation*} a_1=28-13d \end{equation*} \\ a_1=28-13\left(2\right) \\ a_1=28-26 \\ a_1=2 \end{gathered}$

Therefore, the first term of the sequence is 2.