If lim _(x → 1) (x + x² + x³ + ... + xⁿ - n)/(x - 1) = 820,
(n ∈ N) then the value of n is equal to _______.

Solution

Maths sir · 4 steps

Split the -n into n separate -1s, one for each power of x, and every piece becomes a standard limit.

1. Regroup the numerator. At x = 1 the numerator is n - n = 0, so the limit has the form 0/0. Write -n as n copies of -1 and pair one with each power: the numerator becomes (x - 1) + (x² - 1) + … + (xⁿ - 1).
2. Use the standard limit. For each term, lim(x → 1) (xᵏ - 1)/(x - 1) = k. This follows from xᵏ - 1 = (x - 1)(xᵏ⁻¹ + … + x + 1), and the bracket has k terms that each equal 1 at x = 1.

   lim(x → 1) (x + x² + … + xⁿ - n)/(x - 1) = 1 + 2 + … + n = n(n+1)/2

3. Solve for n. Set n(n+1)/2 = 820, so n(n+1) = 1640. Since 40 × 41 = 1640, n = 40. The other root of n² + n - 1640 = 0 is -41, which is not a natural number.
4. So. The value of n is 40.

The idea to keep. When a limit as x → 1 has a sum of powers minus a constant on top, give each power its own -1. Then each piece (xᵏ - 1)/(x - 1) tends to k.

Answer from the NTA final answer key. Solution by PrepWiser.