Question

five terms are in AP the difference between the square of the 5th term and the first term is 192 and the sum of second and The Fourth term is 16 find the 5 terms​

Answer 1

Answer:

The answer is in the above picture

Answer 2

The five terms of  A.P. is 2, 5, 8, 11, 14

Step-by-step explanation:

Given: The difference between the square of the 5th term and the first term is 192 and the sum of second  and the fourth term is 16

To find: The five terms of A.P.

Now as we know

$a_n=a+(n-1)d$ where $a_n$ is the nth term d is the common difference and a if the first term of A.P.

Now as given

The difference between the square of the 5th term and the first term is 192

Therefore

$(a_5)^2-(a)^2= 192\\\\\Rightarrow (a+4d)^2-a^2=192\\\\\Rightarrow a^2+(4d)^2+2 \times a\times 4d -a^2=192\\\\\Rightarrow a^2+16d^2+8ad-a^2=192 \\\\\Rightarrow 16d^2+8ad=192 ------(i)$

Now

The sum of second and the fourth term is 16

$a_2+a_4=16\\\\\Rightarrow a+d+a+3d=16\\\\\Rightarrow 2a+4d=16\\\\\Rightarrow a+2d=8 ------(ii)$

from (ii) we get  a= 8-2d

substituting the value of a in (i) we get

$16d^2+8\times(8-2d)\times d= 192\\\\\Rightarrow 16d^2+64d-16d^2=192\\\\\Rightarrow 64d=192 \\\\\Rightarrow d= \dfrac{192}{64} =3$

Now substitute the value of d = 3 in a= 8-2d

a= 8 - 2 x 3

a= 8-6

a=2

$a=2\\a_2=2+3=5\\a_3=5+3=8\\a_4=8+3=11\\a_5=11+3= 14$

Therefore the A.P. is 2, 5, 8, 11, 14

#Learn more

The sum of how many terms of the AP 10,12,14.....will be 190

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