Maths Help?

Question

he sum of the first five terms of an arithmetic sequence is 40 and the sum of the 
first ten terms of the sequence is 155. What is the sum of the first fifteen terms of the sequence?

Answer 1

Sum of the first n terms of an arithmetic sequence : S_n = (n/2)[2a + (n - 1)d]

Where,

n is number terms in the sequence,

a is the first term of the sequence, and

d is the common difference.

Sum of the first five terms of the arithmetic sequence : S₅ = (5/2)[2a + (5 - 1)d] = 40

(5/2)[2a + 4d] = 40

⇒ a + 2d = 8 → (1)

Sum of the first ten terms of the arithmetic sequence : S₁₀ = (10/2)[2a + (10 - 1)d] = 155

5[2a + 9d] = 155

⇒ 2a + 9d = 31 → (2)

Solve eq (1) & (2) for a and d.

Multiply the first equation by 2.

⇒ 2a + 4d = 16 → (3)

Write the two equations (2) & (3)  in column form, and then subtract them each other, and solve for d.

2a + 9d = 31

2a + 4d = 16

( - )_________

5d = 15

⇒ d = 15/5 = 3.

Substitute the value d = 3 in eq (1), and solve for a.

a + 2 * 3 = 8

a = 8 - 6

⇒ a = 2.

Now, the sum of the first fifteen terms of the arithmetic sequence : S₁₅ = (15/2)[2a + (15 - 1)d].

Substitute the values a = 2 and d = 3.

S₁₅ = (15/2)[2 * 2 + (15 - 1)3]

S₁₅ = (15/2)[4 + (14)3]

S₁₅ = (15/2)[4 + 42]

S₁₅ = (15/2)[46]

S₁₅ = (15)[23]

S₁₅ = 345.

Therefore, the sum of the first fifteen terms of the arithmetic sequence is 345 .