What is the first term of an infinitely decreasing G. P whose sum is 16 and the second term is -0.5?

8 + 6√2  

Step-by-step explanation:

Sum of infinite GP is:

a/(1 - r) = 16 and

The second term is:

ar = -0.5

Eliminate r in the system:

a/(1-r) = 16 ⇒  a = 16 - 16r ⇒ 16r = 16 - a ⇒ r = 1 - a/16ar = -0.5 ⇒ r = -0.5/a

Compare the two equations and solve for a:

1 - a/16 = -0.5/a

Multiply both sides by 16a:

16a - a² = -8a² - 16a = 8 a² - 16a + 64 = 72(a - 8)² = 72a - 8 = ±√72a = 8 ± √72a = 8 + 6√2  a = 8 - 6√2, excluded as it is negative and makes common difference positive and so an increasing function

a = 8 - 6√2 = -0.4852 ⇒ r = -0.5/a = -0.5/-0.4852 = 1.03 > 1