the uneven natural numbers are written in a pattern as shown
row 1: 1
row 2: 3 5 7
row 3: 9 11 13 15 17
row 4: 19 21 23 25 27 29 31
what is the middle number in the 60th row
To solve this problem, we need to find the pattern in the number of elements in each row and the first number in each row.
From observation, we can see that the number of elements in each row increases by 2. In row 1, there is 1 element, in row 2 there are 3 elements, in row 3 there are 5 elements, and so on.
For the first number in each row, we can observe that it follows a pattern as well. Row 1 starts with 1, row 2 starts with 3, row 3 starts with 9, and so on. The first number in each row can be calculated using the formula:
First number in row n = (n-1)^2 + 1
To find the middle number in the 60th row, we need to determine the total number of elements in rows 1 to 60.
Sum of first n odd numbers = n^2
Thus, the number of elements in the 60th row is 60^2 = 3600.
Now, let's find the first number in the 60th row:
First number in row 60 = (60-1)^2 + 1
= 59^2 + 1
= 3481 + 1
= 3482
Since there are 3600 elements in the row, and the middle element is the 1801st element (3600/2 + 1), we'll find the middle number.
Middle number in row 60 = First number in row 60 + 1800
= 3482 + 1800
= 5282
Therefore, the middle number in the 60th row is 5282.
To find the middle number in the 60th row of uneven natural numbers, we need to follow the pattern.
First, let's analyze the pattern:
- Each row starts with an odd number.
- The number of elements in each row follows a sequence: 1, 3, 5, 7, 9, ...
- The difference between consecutive terms in each row is 2.
Now, let's calculate the number of elements in the 60th row:
Number of elements = 1 + 2 + 4 + 6 + 8 + ... (continuing until the 60th term)
Let's break down the calculation:
1 + 2 + 4 + 6 + 8 + ... (continuing until the 60th term)
To find the sum of this series, we use the formula for the sum of an arithmetic series:
Sum = (n/2) * (2a + (n-1)d)
Where:
n = number of terms
a = first term
d = common difference
In this case, since the first term (a) is 1 and the common difference (d) is 2, the formula becomes:
Sum = (n/2) * (2 + (n-1)2) = n^2
Now we can find the value of n (number of terms) when the sum is equal to or greater than 60. Let's solve for n:
n^2 >= 60
n >= √60
n >= 7.75
Since the number of terms must be a whole number, we round up to the nearest whole number:
n = 8
Therefore, the 60th row has 8 elements.
Since the middle number is the (n/2)th term, where n is the number of terms, we can find the middle number as follows:
Middle number = 1 + ((n/2) - 1) * d
Substituting the values:
Middle number = 1 + ((8/2) - 1) * 2
Middle number = 1 + (4-1) * 2
Middle number = 1 + 3 * 2
Middle number = 1 + 6
Middle number = 7
Therefore, the middle number in the 60th row of uneven natural numbers is 7.
To find the middle number in the 60th row of the pattern, we need to understand how the pattern is formed.
Observing the given rows, we can notice that each row starts with an odd number and the number of terms in each row increases linearly. The first row has 1 term, the second row has 3 terms, the third row has 5 terms, and so on.
The nth row of numbers can be represented by the formula:
First Term = (2n - 1) + (n - 1) * 2
Number of Terms = 1 + (n - 1) * 2
Using this formula, we can find the number of terms and the first term in the 60th row:
First Term = (2 * 60 - 1) + (60 - 1) * 2
First Term = 119 + 118
First Term = 237
Number of Terms = 1 + (60 - 1) * 2
Number of Terms = 1 + 59 * 2
Number of Terms = 119
The middle number in a row is simply the average of the first and last term in that row. Since we know the first term and the number of terms, we can find the middle number as follows:
Middle Number = First Term + (Number of Terms - 1) * 2
Middle Number = 237 + (119 - 1) * 2
Middle Number = 237 + 118 * 2
Middle Number = 237 + 236
Middle Number = 473
Therefore, the middle number in the 60th row is 473.