Calculating the Value of n for Equal Summations of Integer Sequences

In mathematics, determining the value of n such that two different summations of integer sequences are equal is a classic problem. This particular problem involves the sum of integers from 1 to n being equal to the sum of integers from n to 49. Understanding and solving such problems is crucial for those interested in arithmetic series and algebraic manipulation.

Introduction to the Problem

The problem presented involves finding the integer n that satisfies the condition where the sum of all integers from 1 to n equals the sum of all integers from n to 49. This requires a deep understanding of the formulas for the sum of the first n integers and the sum of an arithmetic series.

Formulas and Derivations

The sum of the first n integers can be determined using the formula:

[ S_1 frac{n(n 1)}{2} ]

This formula simplifies the process of calculating the sum of a sequence of consecutive integers starting from 1. For the second part of the problem, the sum from n to 49 can be calculated by first finding the sum from 1 to 49, and then subtracting the sum from 1 to n-1:

[ S_2 sum_{kn}^{49} k sum_{k1}^{49} k - sum_{k1}^{n-1} k ]

These sums can be computed explicitly using the following formulas:

[ S_{49} frac{49(49 1)}{2} frac{49 times 50}{2} 1225 ]

[ S_{n-1} frac{(n-1)n}{2} ]

Substituting these into the expression for S_2 gives:

[ S_2 1225 - frac{(n-1)n}{2} ]

Setting the two sums equal to each other:

[ frac{n(n 1)}{2} 1225 - frac{(n-1)n}{2} ]

Multiplying through by 2 to clear the fractions:

[ n(n 1) 2450 - (n-1)n ]

Expanding and simplifying:

[ n^2 n 2450 - n^2 n ]

[ 2n^2 2450 ]

[ n^2 1225 ]

[ n sqrt{1225} ]

[ n 35 ]

Thus, the value of n that satisfies the condition is 35.

Conclusion

By solving the equation derived from the sums of the integer sequences, we have determined that n must be 35 for the sum of integers from 1 to n to equal the sum of integers from n to 49.

References

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