What Is The Following Sum

What Is the Following Sum? Decoding a Fundamental Mathematical Prompt

You’ve likely encountered it in a textbook, on an exam, or in a video tutorial: a problem statement that simply reads, “What is the following sum?” followed by a string of numbers or an expression like 5 + 8 + 11 + ... + 101. At first glance, it seems straightforward—just add everything up. But this deceptively simple prompt is a gateway to a rich world of mathematical thinking, efficiency, and pattern recognition. It’s not merely a test of your ability to perform repetitive addition; it’s an invitation to analyze structure, identify sequences, and apply smarter strategies. Understanding how to approach “the following sum” is a cornerstone skill in algebra, calculus, and beyond, transforming a tedious chore into an elegant intellectual exercise.

The phrase “What is the following sum?” is a standard instructional cue in mathematics. Its core meaning is unambiguous: it asks for the total value obtained by adding all the terms of a presented series or sequence. However, the true educational value lies in how you find that total. The prompt implicitly challenges you to look beyond brute-force calculation. Is the sequence arithmetic (with a constant difference between terms)? Is it geometric (with a constant ratio)? Does it follow a more complex pattern? Your ability to diagnose the underlying rule determines whether you solve the problem in ten seconds or ten minutes (or give up due to tedium). This article will unpack this fundamental concept, moving from basic identification to powerful summation techniques, illustrating why this question is so prevalent and how mastering it builds critical analytical muscles.

Detailed Explanation: From Literal Addition to Strategic Summation

At its most literal, a sum is the result of an addition operation. When presented with a finite list of numbers, the sum is the aggregate total. For example, the sum of 2, 4, 6, 8 is 20. The phrase “the following sum” simply points your attention to the specific list or pattern that comes after it in the text. The complexity arises from how that list is presented.

Often, the sum is not given as an explicit, short list but as a sequence defined by a pattern, sometimes with an ellipsis (...) indicating continuation, and a final term. For instance: 1 + 3 + 5 + ... + 99. Here, you are not meant to manually add all 50 odd numbers. Instead, you must recognize this as an arithmetic series—a sequence where each term increases by a constant common difference (d=2). The first term (a₁) is 1, and the last term (aₙ) is 99. The genius of mathematics provides us with formulas to find the sum of such series without exhaustive addition.

The general formula for the sum Sₙ of the first n terms of an arithmetic series is: Sₙ = n/2 * (a₁ + aₙ) or equivalently, Sₙ = n/2 * [2a₁ + (n-1)d]

This formula works because you can pair the first and last terms, the second and second-last terms, and so on. Each pair sums to the same value (a₁ + aₙ), and there are n/2 such pairs. This pairing insight, often attributed to the young Carl Friedrich Gauss, is the key that unlocks the efficiency of the method.

In contrast, a geometric series has a constant common ratio (r) between successive terms. For example: 3 + 6 + 12 + 24 + ... + 768. Here, r=2. The sum formula is different: Sₙ = a₁ * (1 - rⁿ) / (1 - r) for r ≠ 1. This formula derives from a clever algebraic manipulation of the series itself. Recognizing the type of series is the mandatory first step in solving “what is the following sum?” efficiently and correctly.

Step-by-Step Breakdown: A Strategic Approach

When faced with the prompt “What is the following sum?”, follow this logical diagnostic sequence:

1. Identify the Pattern: Examine the given terms. Is there a constant difference? Subtract the second term from the first, the third from the second. If the result is always the same, you have an arithmetic sequence. If there is a constant ratio (divide successive terms), you have a geometric sequence. If the differences themselves change consistently (e.g., second differences are constant), you might be dealing with a quadratic sequence, which has its own summation methods involving polynomial expressions.

2. Extract Key Parameters: Once the sequence type is known, label the essential components:

   • a₁: The first term.
   • aₙ: The last term (if explicitly given or easily found).
   • d or r: The common difference or ratio.
   • n: The number of terms. This is often the trickiest part to determine. You must derive n from the pattern and the last term. For an arithmetic sequence, use the formula for the nth term: aₙ = a₁ + (n-1)d. Solve for n.

3. Select and Apply the Correct Formula: Match your identified sequence type to its summation formula. Plug in the values for a₁, aₙ (or d/r), and n. Perform the arithmetic carefully.

4. Verify with a Small Sample: If unsure, calculate the sum of the first 3 or 4 terms manually and compare it to the result from your formula applied to just those terms. This sanity check catches errors in identifying n or the common difference/ratio.

Example Walkthrough: What is the following sum? 7 + 12 + 17 + ... + 102

• Step 1: Differences: 12-7=5, 17-12=5. Arithmetic with d=5.
• Step 2: a₁=7, aₙ=102, d=5. Find n: 102 = 7 + (n-1)*5 ->