Question

the terms of an arithmetic sequence with common difference 4 are natural numbers
a)If the sum of reciprocals of two consecutive terms of this sequence is 4/15,find those terms?​

Answer 1

Given :-

• Common difference of AP is 4
• sum of reciprocal of the consecutive terms is 4/15

To Find

• We have to find the terms of the AP

Solution:-

Let the first term of AP be a and 2nd term be a+4

Now ,

$\underline{\bigstar{\sf\ \ According \ to\ Question:-}}$

$:\implies\sf\ \dfrac{1}{a}+\dfrac{1}{a+4}=\dfrac{4}{15}\\ \\ \\ \\ :\implies\sf\ \dfrac{a+4+a}{a(a+4)}=\dfrac{4}{15}\\ \\ \\ \\ :\implies\sf\ \dfrac{2a+4}{a^2+4a}=\dfrac{4}{15}\\ \\ \\ \\ :\implies\sf\ 15(2a+4)=4(a^2+4a)\\ \\ \\ \\ :\implies\sf\ 30a+60=4a^2+16a\\ \\ \\ \\ :\implies\sf\ 4a^2-14a-60=0\\ \\ \\ \\ :\implies\sf\ 2(2a^2-7a-30)=0\\ \\ \\ \\ :\implies\sf\ 2a^2-7a-30= \dfrac{0}{2}\\ \\ \\ \\ :\implies\sf\ 2a^2-(12-5)a-30=0\\ \\ \\ \\ :\implies\sf\ 2a^2-12a+5a-30=0\\ \\ \\ \\ :\implies\sf\ 2a(a-6)+5(a-6)=0\\ \\ \\ \\ :\implies\sf\ (a-6)(2a+5)=0\\ \\ \\ \\ :\implies\sf\ a=6\ \ or\ a= \dfrac{-5}{2}$

• But the term of sequence is natural numbers so,. the value of a cannot be negative or fraction

$\therefore\sf\ a=6 \\ \\ \sf\ a_2= 6+4=10$

•°• the sequence be 6,10,14,18.....

Answer 2

Answer:

$\huge{\bf{\underline{\red{Answer :}}}}$

★The terms of an arithmetic sequence with common difference 4 are natural numbers.

(a) Let x be a term of the sequence and y be the next term of the sequence.

____________________________

Common difference = yx

$\:\::\:\\:$4 = yx

$\:\:\:\:\:$y = X + 4

So, The next term is x + 4

B) Let x and x + 4 be the two consecutive terms of sequence such that sum of their $\frac{4}{5}$

$\Large\frac{1}{x} + \frac{1}{x + 4} = \frac{4}{15}$

$\Large\frac{2x + 4}{ {x}^{2} + 4 } = \frac{4}{5}$

• 30x + 60 = 4x² + 16x

• 4x² - 14x - 60 = 0

• 4x² - 24x + 10x - 60 = 0

• 4x (x−6) + 10 (x−6) = 0

• (4x+10) (x−6) = 0

$x = \frac{ - 5}{2} or \: x = 6$

__________________________

But terms of the sequence are natural numbers.

$\boxed{\red{∴ x=6\:and\:x+4 = 10}}$

The two terms of the sequence are 6 and 10.