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

To find the number of terms in the series \(3 + 11 + 19 + \ldots = 1580\), we first need to identify the pattern. The series is an arithmetic sequence where the first term \(a = 3\) and the common difference \(d = 8\) (since \(11 - 3 = 8\)). The \(n\)-th term can be expressed as: \[ a_n = a + (n-1)d = 3 + (n-1) \times 8 = 8n - 5 \] To find \(n\), we set the sum of the first \(n\) terms equal to \(1580\): \[ S_n = \frac{n}{2} \times (a + a_n) = \frac{n}{2} \times (3 + (8n - 5)) = \frac{n}{2} \times (8n - 2) = 4n^2 - n \] Setting this equal to \(1580\): \[ 4n^2 - n - 1580 = 0 \] Now, we can solve this quadratic equation using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -1\), and \(c = -1580\): \[ b^2 - 4ac = (-1)^2 - 4 \times 4 \times (-1580) = 1 + 25280 = 25281 \] Calculating \(n\): \[ n = \frac{1 \pm \sqrt{25281}}{8} \] Calculating \(\sqrt{25281}\): \(\sqrt{25281} \approx 159.1\), so: \[ n = \frac{1 + 159}{8} \approx 20 \] Verifying \(n = 20\): Inserting \(n = 20\) back into \(S_n\): \[ S_{20} = 4(20)^2 - 20 = 1600 - 20 = 1580 \] So, the number of terms in the series is **20**. Now let's dive into some additional knowledge! The arithmetic sequence is one of the earliest forms of sequences studied by mathematicians, dating back to ancient civilizations. The ancient Egyptians and Babylonians employed these sequences for trade and land division. It reflects a universal concept, showing how people connected with numbers thousands of years ago. In the real world, understanding arithmetic sequences can help you in budgeting, project planning, and even in understanding patterns in nature, like the growth of trees or the arrangement of leaves. When you spot a repeating pattern, like your favorite song following a specific beat, you might just be encountering an arithmetic sequence in musical form!