Question

Solve for x: 1 + 4 + 7 + 10 + ... + x = 287.

Answer 1

1 + 4 + 7 + 10 + ... + x = 287

Here, a = 1, d = 4 – 1 = 3, n = x

l = x = a = (n – 1)d = 1 + (n – 1) × 3

⇒ x – 1 = (n – 1)d

S_n = `n/(2)[2a + (n - 1)d]`

287 = `n/(2)[2 xx 1 + (n - 1)3]`

574 = n(2 – 3n – 3)

⇒ 3n² – n – 574 = 0

⇒ 3n² – 42n + 41n – 574 = 0

⇒ 3n(n – 14) + 41(n – 14) = 0

⇒ (n – 14)(3n + 41) = 0

Either n – 14 = 0

Then n = 14

or

3n + 41 = 0,

Then 3n = –41

⇒ n = `(-41)/(3)`

Which is not possible being negative.

∴ n = 14

Now, x = a + (n – 1)d

= 1 + (14 – 1) × 3

= 1 + 13 × 3

= 1 + 39

= 40

∴ x = 40