Question

If given $ 5 $ times the fifth term of an A.P. is equal to $ 8 $ times its eight term, show that its $ 13th $ term is zero.

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: $ {t_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 fifth, eighth and thirteen terms using the standard formula and then will place value as per the given condition.

Complete step-by-step answer:
Now, fifth term of an arithmetic progression –
 $ {t_5} = a + (5 - 1)d $
Simplify the above equation –
 $ \Rightarrow {t_5} = a + 4d $ .... (A)
Similarly,
 $ {t_8} = a + (8 - 1)d $
Simplify the above equation –
 $ \Rightarrow {t_8} = a + 7d $ .... (B)
And –
 $ {t_{13}} = a + (13 - 1)d $
Simplify the above equation –
 $ \Rightarrow {t_{13}} = a + 12d $ .... (C)
Given that five times the fifth term is equal to eight times the eighth term.
 $ \Rightarrow 5.{t_5} = 8.{t_8} $
Place values in the above equation from (A) and (B)
 $ \Rightarrow 5(a + 4d) = 8(a + 7d) $
Open the brackets and simplify –
 $ \Rightarrow 5a + 20d = 8a + 56d $
The above equation can be re-written as –
 $ \Rightarrow 8a + 56d = 5a + 20d $
Take all the terms on the left hand side of the equation. Also, when you move a term from one side of the equation to another, the sign also changes. Positive terms become negative and vice-versa.
 $ \Rightarrow 8a + 56d - 5a - 20d = 0 $
Make the pair of like terms –
 $ \Rightarrow \underline {8a - 5a} + \underline {56d - 20d} = 0 $
Simplify –
 $ \Rightarrow 3a + 36d = 0 $
Take common multiple from the above equation –
 $ \Rightarrow 3(a + 12d) = 0 $
 $ \Rightarrow a + 12d = 0 $ ....(D)
We can observe that equation (C) and (D) are equal.
Hence, proved that - five times the fifth term is equal to eight times the eighth term which is the thirteen term.
So, the correct answer is “ $a + 12d = 0 $”.

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.