Question:** A plant biologist discovers a growth pattern in a drought-resistant crop where the number of new root branches per week follows the sequence: 2, 5, 8, 11, ... What is the sum of the number of root branches produced in the first 30 weeks?

["Understanding the Growth Pattern: How a Drought-Resistant Plant Develops Root Branches", "A pioneering plant biologist has uncovered a fascinating growth pattern in a recently studied drought-resistant crop. Through careful observation, they discovered that the number of new root branches produced each week follows a distinct arithmetic sequence: 2, 5, 8, 11, ... Understanding this sequence offers valuable insights into the plant’s adaptive strategies and opens new possibilities for agricultural innovation in arid environments. But beyond its biological significance, this pattern also presents a compelling mathematical challenge—how can we calculate the total root branch production over the first 30 weeks?", "### What Is the Growth Pattern?", "The sequence 2, 5, 8, 11, ... is an arithmetic progression, where each term increases by a constant difference.", "- First term (a₁): 2 (root branches in week 1)\n- Common difference (d): 3 (each week sees 3 more root branches than the previous week)", "The general formula for the n-th term of an arithmetic sequence is:\n[ a_n = a_1 + (n - 1)d ]\nSubstituting the values:\n[ a_n = 2 + (n - 1) \cdot 3 = 3n - 1 ]", "So, the number of new root branches in week n is given by ( a_n = 3n - 1 ).", "### Calculating the Total Root Branch Production", "To find the total number of root branches over 30 weeks, we need the sum of the first 30 terms of this sequence, denoted as ( S_{30} ).", "For arithmetic sequences, the sum is calculated using the formula:\n[ S_n = \frac{n}{2}(a_1 + a_n) ]", "We already have:\n- ( n = 30 )\n- ( a_1 = 2 )\n- ( a_{30} = 3(30) - 1 = 90 - 1 = 89 )", "Substituting into the sum formula:\n[ S_{30} = \frac{30}{2}(2 + 89) = 15 \ imes 91 = 1365 ]", "### Why This Matters", "The total of 1,365 root branches produced in the first 30 weeks illustrates the efficient resource allocation of this drought-adapted species. By extending lateral roots weekly in a steady, predictable pattern, the plant maximizes soil exploration without excessive energy cost—key traits for survival in parched regions.", "From a plant biologist’s perspective, recognizing such patterns enables better predictions of crop resilience and informs breeding programs aimed at enhancing drought tolerance. For agricultural researchers, arithmetic sequences like this serve as models for understanding growth dynamics in hardy crops, contributing to food security in a changing climate.", "### Conclusion", "The discovery of the root branch sequence in this drought-resistant plant reveals not only a key piece of its adaptive biology but also a clear mathematical pathway to quantifying its growth. With a total of 1,365 root branches developing in the first 30 weeks, this finding underscores the power of combining botanical science with mathematical modeling. Such interdisciplinary insights are critical for nurturing the next generation of resilient crops.", "Keywords: drought-resistant crop, root branch growth, arithmetic sequence in plants, plant biologist discovery, 30 week growth sum, plant adaptation, agricultural science, root development pattern, sum of arithmetic sequence."]