# Problem of the Week Problem D and Solution Sum New Year!

## Problem

The positive integers are written consecutively in rows, with seven integers in each row. That is, the first row contains the integers $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, and $$7$$. The second row contains the integers $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, and $$14$$. The third row contains the integers $$15$$, $$16$$, $$17$$, $$18$$, $$19$$, $$20$$, and $$21$$, and so on.

The row sum of a row is the sum of the integers in the row. For example, the row sum of the first row is $$1+2+3+4+5+6+7=28$$.

Determine the numbers in the row that has a row sum closest to $$2024$$.

## Solution

Solution 1

The last number in row $$1$$ is $$7$$, the last number in row $$2$$ is $$14$$, and the last number in row $$3$$ is $$21$$. Observe that the last number in each row is a multiple of $$7$$. Furthermore, the last number in row $$n$$ is $$7n$$. Since the last number in row $$n$$ is $$7n$$, the six preceding numbers in the row are $$7n-1$$, $$7n-2$$, $$7n-3$$, $$7n-4$$, $$7n-5$$, and $$7n-6$$.

The sum of the numbers in row $$n$$ is $(7n-6)+(7n-5)+(7n-4)+(7n-3)+(7n-2)+(7n-1)+7n= 49n-21$ We want to find the integer value of $$n$$ so that $$49n-21$$ is as close to $$2024$$ as possible.

\begin{aligned} 49n-21&=2024\\ 49n&=2045\\ n&\approx 41.7 \end{aligned}

The closest integer to $$41.7$$ is $$42$$. The row sum of row $$42$$ is $$49n-21=49(42)-21=2037$$. The last number in row $$42$$ is $$7\times 42=294$$. The seven integers in the row $$42$$ are $$288$$, $$289$$, $$290$$, $$291$$, $$292$$, $$293$$, and $$294$$. Row $$41$$ contains the integers $$281$$, $$282$$, $$283$$, $$284$$, $$285$$, $$286$$, and $$287$$, and has row sum equal to $$1988$$. This row sum is farther from $$2024$$ than $$2037$$, the row sum of row $$42$$.

Therefore, row $$42$$ has the row sum closest to $$2024$$. This row contains the integers $$288$$, $$289$$, $$290$$, $$291$$, $$292$$, $$293$$, and $$294$$.

The second solution approaches the problem by establishing a linear relationship.

Solution 2

Let $$x$$ represent the row number and $$y$$ represent the sum of the integers in the row. Observe that the seventh integer in any row is a multiple of $$7$$. In fact, the seventh integer in any row is $$7$$ times the row number or $$7x$$. The following table of values summarizes the row sums for the first three rows.

Row Number ($$x$$) Row Sum ($$y$$)
$$1$$ $$28$$
$$2$$ $$77$$
$$3$$ $$126$$

Notice that the $$y$$ values increase by $$49$$ as the $$x$$ values increase by $$1$$. We will verify that this is true. In Solution $$1$$ we saw that the sum of the numbers in row $$n$$ is $$49n-21$$. Therefore, the sum of the numbers in row $$x$$ is $$49x - 21$$ and the sum of the numbers in row $$(x+1)$$ is $$49(x+1) - 21 = (49x -21) + 49$$. Thus, the $$y$$ values increase by $$49$$ as the $$x$$ values increase by $$1$$. This tells us that the sum of the fourth row should be $$126+49=175$$. We can verify this by adding $$22+23+24+25+26+27+28$$, the numbers in the fourth row. The sum is indeed $$175$$.

As the values of $$x$$ increase by $$1$$, the values of $$y$$ increase by $$49$$. The relation is linear. The slope is $$\dfrac{\Delta y}{\Delta x}=\dfrac{49}{1}=49$$. Substituting $$x=1$$, $$y=28$$, $$m=49$$ into the equation $$y=mx+b$$, we get \begin{aligned} 28&=49(1)+b\\ -21&=b \end{aligned}

Thus, the equation of the line which passes through the points in the relation is $$y=49x-21$$. Note that $$x$$ and $$y$$ are positive integers. We want to find the value of $$x$$, the row number, so that the value of $$y$$, the row sum, is as close to $$2024$$ as possible. \begin{aligned} 49x-21&=2024\\ 49x&=2045\\ x&\approx 41.7 \end{aligned} The closest integer to $$41.7$$ is $$42$$. When $$x=42$$, the row sum is $$y=49(42)-21=2037$$. The row sum when $$x=41$$ is $$y=49(41)-21=1988$$. The row sum $$2037$$ is closer to $$2024$$ than the row sum $$1988$$.

Therefore, row $$42$$ has the row sum closest to $$2024$$. The seventh number in row $$42$$ is $$7\times 42=294$$. Thus, this row contains the integers $$288$$, $$289$$, $$290$$, $$291$$, $$292$$, $$293$$, and $$294$$.