The sum of the first n natural numbers is given by the formula n(n + 1)/2 = 255. Solving n(n + 1) = 510, we find n = 16.

The sum of the first n natural numbers is given by the formula n(n + 1)/2. Setting this equal to 2550, we have n(n + 1) = 5100. Solving the quadratic equation n^2 + n - 5100 = 0 gives n = 50.

Let the other number be x. Then, 9 + x = 15. Thus, x = 15 - 9 = 6.

Let the numbers be x and y. We have x + y = 30 and x - y = 10. Solving these gives x = 20 and y = 10.

To find x, subtract 5 from both sides: 3x = 20 - 5 = 15. Then divide by 3: x = 15 / 3 = 5.

'Costs an arm and a leg' means something is very expensive.

'Hit the nail on the head' means to describe exactly what is causing a situation or problem.

'Piece of cake' means something that is very easy to do.

'The ball is in your court' means it is now your responsibility to take action or make a decision.

'Barking up the wrong tree' means to pursue a mistaken or misguided course of action.

'Burn the midnight oil' means to stay up late working or studying.

'Under the weather' means feeling ill or unwell.

'Diligent' means showing care and effort in one's work, making 'industrious' the correct synonym.

'Hasty' means done with excessive speed or urgency; thus, the synonym is 'quick'.

'Meticulous' means showing great attention to detail; thorough.

'Contract' is the opposite of 'expand', meaning to decrease in size.

'Diminish' is the opposite of 'increase', which means to make smaller or reduce.

'Quiet' is the opposite of 'noisy', referring to a lack of sound.

'Complicated' is the opposite of 'simple', meaning something that is difficult to understand.

'Fail' is the opposite of 'succeed', indicating a lack of success.

'Defeat' is the opposite of 'victory', indicating loss.

'Obscure' means not visible or hard to see, which is the opposite of 'visible'.

'A blessing in disguise' refers to something that seems bad at first but results in something good.

'Costs an arm and a leg' means something is very expensive.

'Once in a blue moon' means something that happens very rarely.

'Spill the beans' means to disclose confidential information or secrets.

'The ball is in your court' means it is your responsibility to take action or make a decision.

'To break the ice' means to initiate conversation in a social situation, especially among strangers.

'Let the cat out of the bag' means to accidentally reveal a secret.

Adding 4 to both sides gives 2x = 14. Dividing by 2 gives x = 7.