Question

How do you write the first five terms of the arithmetic sequence given $ {a_8} = 26,\;{{\text{a}}_{12}} = 42? $

Answer 1

Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is: $ {a_n} = a + (n - 1)d $
Where $ {t_n} = $ nth term in the AP
  $ a = $ First term of AP
 $ d = $ Common difference in the series
 $ n = $ Number of terms in the AP
Here we will find the eighth and twelfth value in the standard formula and then will simplify both the equations and then will find the first five terms of Arithmetic Sequence.

Complete step-by-step answer:
Now, Eighth term of an arithmetic progression –
 $ {a_8} = a + (8 - 1)d $
Simplify the above equation and place the given value –
 $ \Rightarrow 26 = a + 7d $ .... (A)
Similarly, twelfth term can be given by -
 $ {a_{12}} = a + (12 - 1)d $
Simplify the above equation and place the given data –
 $ \Rightarrow 42 = a + 11d $ .... (B)
Simplify the above equations A and B –
Subtract equation (A) from equation (B)
 $ \Rightarrow 42 - 26 = 11d - 7d $
Simplify the above equation –
 $ \Rightarrow 16 = 4d $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow d = 4 $ ….. (C)
Place the above value in the equation (A)
 $ \Rightarrow 26 = a + 7(4) $
Simplify the above equation-
 $ \Rightarrow a = ( - 2) $ …… (D)
Now, First five terms of the arithmetic sequence can be given by –
 $ a,{\text{ a + d, a + 2d, a + 3d, a + 4d}} $
Placing the values gives –
 $ ( - 2),\;{\text{2, 6, 10, 14}} $
This is the required solution.
So, the correct answer is “ $ ( - 2),\;{\text{2, 6, 10, 14}} $”.

Note: Be careful about the signs of the terms while simplification. When the terms are moved from one side to another, Sign of the term is also changed. Positive terms become negative and vice-versa. When you add one positive and one negative, you have to subtract and give a sign of the bigger number.