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

PLEASE HELP ME! If an arithmetic sequence has a4 = 107 and a8 = 191, what is a₁? (I-example-1

2 Answers

14 votes

Answer:


\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}

User Federkun
by
8.3k points
12 votes

Answer:


\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.

User Vriesdemichael
by
7.6k points

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