Question

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

Answer 1

• 8 + 6√2

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/16
• ar = -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 = -8
• a² - 16a = 8
• a² - 16a + 64 = 72
• (a - 8)² = 72
• a - 8 = ±√72
• a = 8 ± √72
• a = 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