Question

the 1st, 5th and 10th terms of a linear sequence are GP. if the sum of the 2nd and 8th term of the linear sequence is 30 find the first term

Answer 1

• 12 or 15

Explanation:

Let the first term is a and common difference is d.

Given:

• a, a₅ and a₁₀ form a GP ⇒ a₅/a = a₁₀/a₅
• a₂ + a₈ = 30
• a = ?

Use the equation of nth term:

• a₅ = a + 4d
• a₁₀ = a + 9d
• a₂ = a + d
• a₈ = a + 7d

According to the given we have:

• (a + 4d)² = a(a + 9d)
• a² + 8ad + 16d² = a² + 9ad
• 16d² = ad
• d(16d - a) = 0
• 1. d = 0

or

• 2. a = 16d

Now, use the equation of sum:

• a + d + a + 7d = 30
• 2a + 8d = 30
• a + 4d = 15
• a = 15 - 4d

Compare the value of a and find d:

1. d = 0

• a = 15 - 4*0 = 15, in this case all the terms are same

2. a = 16d

• 15 - 4d = 16d
• 20d = 15
• d = 15/20
• d = 0.75

Find a:

• a = 16*0.75
• a = 12