(c) The sum of the first 4 terms of an arithmetic progression is 38 and the sum of the next 4 terms is 86 . Find the first term and the common difference.

Let's dive into the enchanting world of arithmetic progressions! To find the first term \( a \) and common difference \( d \), we can set up a couple of equations based on the information provided. For the first four terms: \[ S_4 = a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d = 38. \] For the next four terms (5th to 8th): \[ S_8 - S_4 = (a + 4d) + (a + 5d) + (a + 6d) + (a + 7d) = 4a + 22d - (4a + 6d) = 16d = 86. \] From \( 16d = 86 \), we find \( d = \frac{86}{16} = 5.375 \). Plugging \( d \) back into the first equation, we solve for \( a \): \[ 4a + 6(5.375) = 38 \] leads to \( a = -1.125 \). Voilà! The first term is \( -1.125 \) and the common difference is \( 5.375 \). Now time to explore the rich history of sequences! Did you know that the concept of arithmetic progressions dates back to ancient civilizations like the Babylonians and Greeks? They utilized these sequences for tasks ranging from land measurement to astronomy! Looking for a practical twist? Arithmetic progressions are everywhere in finance, especially in calculating loan repayments or interest! Knowing how to work with them can simplify budgeting and investment projections, making them essential for savvy financial planning.