Question

PLEASE HELP ME! If an arithmetic sequence has a4 = 107 and a8 = 191, what is a₁? (I need the steps too please). ​

Answer 1

`\huge\boxed{\bf\:a_(1) = 44}`

Explanation:

In the given arithmetic series,

• 
  `a_(4) = 107`

• 
  `a_(8) = 191`

• 
  `a_(1) = ?`

We know that,

`\boxed{a_(n) = a + (n - 1)d}`

Therefore,

`a_(4) = a + 3d = 107 -----(1)\\a_(8) = a + 7d = 191 -----(2)`

By solving the two equations (1) & (2)....

`\:\:\:\:a + 7d = 191 \\-\underline{a + 3d = 107 }\\ \underline{\underline{\:\:\:\:\:\:\:\:\:\:\:\:\:4d = 84\:\:\:\:\:\:\:\:}}\\\\\\\\4d = 84\\d = 84 / 4\\\boxed{d = 21}`

Now, we have the value of the common difference (d). To find
`a_(1)` or a, let's substitute the value of 'd' in (1).

`a + 3d =107\\a + 3(21)=107\\a + 63 = 107\\a = 107 - 63\\\boxed{\bf\:a_(1) = 44}`

`\rule{150pt}{2pt}`

Answer 2

`\displaystyle a_1 = 44`

Explanation:

Recall the direct formula for an arithmetic sequence:

`\displaystyle a_n = a_1 + d(n-1)`

Where d is the common difference.

Therefore, we can write the following two equations:

`\displaystyle \left \{ {a_4 = 107 = a_1 + d(4-1)\atop {a_8 = 191 = a_1 + d(8-1)}} \right.`

Solve the system. Simplifying yields:

`\displaystyle 107 = a_1 + 3d\text{ and } 191 = a_1 + 7d`

Subtracting the two equations into each other yields:

`\displaystyle \begin{aligned} (191) - (107) & = (a_1 + 7d) - (a_1 + 3d) \\ \\ 84 & = 4d \\ \\ d & = 21 \end{aligned}`

Using either equation, solve for the initial term:

`\displaystyle \begin{aligned} 107 & = a_1 + 3(21) \\ \\ a_1 & = 107-63 \\ \\ & = 44\end{aligned}`

In conclusion, the initial term is 44.