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How do you write the first six terms of an arithmetic sequence if the first term ${a_1}$, is $ - 4$, and the common difference, $d$, is $10$?

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Last updated date: 22nd Jun 2024
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Answer
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Hint: Here, we are given the first term and the common difference of an arithmetic sequence. Therefore, we need to use the formula for finding the $nth$term of the arithmetic sequence. By this formula and the given information, we can obtain a linear equation with variable $n$in which we can obtain the terms by putting the value of $n$.

Formula used:${a_n} = {a_1} + \left( {n - 1} \right)d$, where, ${a_n}$is the $nth$ term of the arithmetic sequence, ${a_1}$ is the first term of the arithmetic sequence and $d$is the common difference between two consecutive terms in the arithmetic sequence

Complete step-by-step solution:
We know that for the arithmetic sequence,
${a_n} = {a_1} + \left( {n - 1} \right)d$
We are given that the first term ${a_1} = - 4$and the common difference $d = 10$.
$
   \Rightarrow {a_n} = - 4 + \left( {n - 1} \right)10 \\
   \Rightarrow {a_n} = - 4 + 10n - 10 \\
   \Rightarrow {a_n} = 10n - 14 \\
 $
We are asked to find the first six terms of the arithmetic sequence.
The first term is already given to us which is ${a_1} = - 4$.
We can find the second term by putting the value of $n = 2$ in the equation ${a_n} = 10n - 14$.
$ \Rightarrow {a_2} = 10\left( 2 \right) - 14 = 20 - 14 = 6$
We can find the second term by putting the value of $n = 3$ in the equation ${a_n} = 10n - 14$.
$ \Rightarrow {a_3} = 10\left( 3 \right) - 14 = 30 - 14 = 16$
We can find the second term by putting the value of $n = 4$ in the equation ${a_n} = 10n - 14$.
$ \Rightarrow {a_4} = 10\left( 4 \right) - 14 = 40 - 14 = 26$
We can find the second term by putting the value of $n = 5$ in the equation ${a_n} = 10n - 14$.
$ \Rightarrow {a_5} = 10\left( 5 \right) - 14 = 50 - 14 = 36$
We can find the second term by putting the value of $n = 6$ in the equation ${a_n} = 10n - 14$.
$ \Rightarrow {a_6} = 10\left( 6 \right) - 14 = 60 - 14 = 46$
Thus, the first six terms of the arithmetic sequence are $ - 4,6,16,26,36,46$.

Note: In this type of question where the first term and the common difference of an arithmetic sequence is given, we can also obtain the first few terms by simply adding the common difference to the consecutive terms. For example, here the first term is given as $ - 4$. When we simply add the common difference $10$, we get our second term $6$. Thus, by repeating this four more times, we get the first six terms of the arithmetic sequence.