Question

five terms are in AP the sum is 15 the ratio of the product of extremes to the product of the second and The Fourth term is 55: 7 find the AP​

Answer 1

Answer:

AP  -5 , -1 , 3 , 7 , 11 or  11 , 7 , 3 , -1 , -5

Step-by-step explanation:

Given: AP has 5 terms

           Sum of 5 terms = 15

           Ratio of product of extremes to product of 2nd and 4th term = 55 : 7

To find: AP

$S_n=\frac{n}{2}(2a+(n-1)d)\\\\\implies S_5=\frac{5}{2}(2a+(5-1)d)\\\\15=\frac{5}{2}(2a+4d)\\\\3=a+2d\\\\\implies 3rd\,term = 3\\a=3-2d.........(1)$

$\frac{a\times a_5}{a_2\times a_4}=\frac{55}{7}\\\\\frac{a\times(a+4d)}{(a+d)\times(a+3d)}=\frac{55}{7}\\\\\frac{a^2+4ad}{a^+4ad+3d^2}=\frac{55}{7}\\\\7a^2+28ad=55a^+220ad+165d^2\\48a^2+192ad+165d^2=0\\16a^2+64ad+55d^2=0\\from\.(1)\\\\16(3-2d)^2+64(3-2d)+55d^2=0\\144+64d^-192d+192d-128d^2+55d^2=0\\144-9d^2=0\\d^2=16\\d=\pm4$

when d = 4

a = 3 - 2d = 3 - 8 = -5

AP = a , a+d , a+2d , a+3d , a+4d = -5 , -1 , 3 , 7 , 11

when d = -4

a = 3 - 2d = 3 + 8 = 11

AP = a , a+d , a+2d , a+3d , a+4d = 11 , 7 , 3 , -1 , -5

Therefore, AP  -5 , -1 , 3 , 7 , 11 or  11 , 7 , 3 , -1 , -5