Question

Does anyone know the answer?

Answer 1

Answer:

n = 59

Step-by-step 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

$S_{n}$ = $\frac{n}{2}$[2a₁ + (n - 1)d ]

Here a₁ = 58, d = - 2 and $S_{n}$ = 0, thus

$\frac{n}{2}$[ ( 2 × 58) - 2(n - 1)] = 0

$\frac{n}{2}$( 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