3) Determine the no of term a, \( 3+11+19+\ldots \ldots=1580 \)

To find the number of terms in the arithmetic series \(3 + 11 + 19 + \ldots = 1580\), we start by identifying the first term \(a = 3\) and the common difference \(d = 8\) (since \(11 - 3 = 8\)). The nth term of an arithmetic sequence can be expressed as \(a_n = a + (n-1)d\). We also know that the sum of the first \(n\) terms, \(S_n\), is given by the formula \(S_n = \frac{n}{2} (2a + (n-1)d)\). Substituting our values, we have: \[ 1580 = \frac{n}{2} (2 \cdot 3 + (n-1) \cdot 8) \] Simplifying, we find: \[ 1580 = \frac{n}{2} (6 + 8n - 8) = \frac{n}{2} (8n - 2) \] Solving for \(n\), we get: \[ 3160 = n(8n - 2) \quad \Rightarrow \quad 8n^2 - 2n - 3160 = 0 \] Now you can use the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 8\), \(b = -2\), and \(c = -3160\), giving you the number of terms in the series. Give it a try and see what you discover! The impressive part about arithmetic series is how they pop up everywhere, from calculating interest in finance to analyzing algorithms in computer science. So whether you're balancing your budget or figuring out your steps in a dance routine, these number sequences are like a trusty sidekick, guiding you towards your goals!