Question

Does anyone know the answer?

Answer 1

n = 59

Explanation:

Since the difference between consecutive terms is constant, then the sequence is arithmetic.

The n th term of an arithmetic sequence is

`a_(n)` = a₁ + (n - 1)d

Given

`a_(10)` = 40, then

a₁ + 9d = 40 → (1)

Given

`a_(20)` = 20, then

a₁ + 19d = 20 → (2)

Subtract (1) from (2) term by term

10d = - 20 ( divide both sides by 10 )

d = - 2

Substitute d = - 2 in (1)

a₁ - 18 = 40 ( add 18 to both sides )

a₁ = 58

The sum to n terms of an arithmetic sequence is

`S_(n)` =
`(n)/(2)`[2a₁ + (n - 1)d ]

Here a₁ = 58, d = - 2 and
`S_(n)` = 0, thus

`(n)/(2)`[ ( 2 × 58) - 2(n - 1)] = 0

`(n)/(2)`( 116 - 2n + 2) = 0

Multiply through by 2

n(118 - 2n) = 0

Equate each factor to zero and solve for n

n = 0

118 - 2n = 0 ⇒ - 2n = - 118 ⇒ n = 59

However, n > 0 ⇒ n = 59