The arithmetic sequence $2,4,6,8,10, \ldots$ represents the set of even natural numbers.
What is the 100th even natural number?
$a_{100}=$
What is the sum of the first 100 even natural numbers?
$S_{100}=$
Complete the sentence below with less than, greater than, or equal to.
The sum of the first 100 even natural numbers is
$\square$ the sum of the first 101 odd natural numbers.
Answer (2)
Find the 100th even natural number using the formula a n = a 1 + ( n − 1 ) d : a 100 = 200 .
Calculate the sum of the first 100 even natural numbers using the formula S n = 2 n ( a 1 + a n ) : S 100 = 10100 .
Determine the sum of the first 101 odd natural numbers: S 101 = 10201 .
Compare the two sums: The sum of the first 100 even natural numbers is less than the sum of the first 101 odd natural numbers. l ess t han
Explanation
Understanding the Problem We are given an arithmetic sequence of even natural numbers: 2 , 4 , 6 , 8 , 10 , … . We need to find the 100th term, the sum of the first 100 terms, and compare the sum of the first 100 even natural numbers with the sum of the first 101 odd natural numbers.
Finding the 100th Term The first term of the sequence is a 1 = 2 , and the common difference is d = 2 . To find the 100th term, we use the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d . Substituting n = 100 , a 1 = 2 , and d = 2 , we get: a 100 = 2 + ( 100 − 1 ) × 2 = 2 + 99 × 2 = 2 + 198 = 200.
Finding the Sum of the First 100 Terms Now, we need to find the sum of the first 100 terms. We use the formula for the sum of the first n terms of an arithmetic sequence: S n = 2 n ( a 1 + a n ) . Substituting n = 100 , a 1 = 2 , and a 100 = 200 , we get: S 100 = 2 100 ( 2 + 200 ) = 50 × 202 = 10100.
Finding the Sum of the First 101 Odd Natural Numbers Next, we need to find the sum of the first 101 odd natural numbers. The sequence of odd natural numbers is 1 , 3 , 5 , 7 , … . This is also an arithmetic sequence with first term a 1 = 1 and common difference d = 2 . The nth term is given by a n = a 1 + ( n − 1 ) d . So the 101th term is a 101 = 1 + ( 101 − 1 ) × 2 = 1 + 100 × 2 = 1 + 200 = 201 . The sum of the first 101 odd natural numbers is S 101 = 2 101 ( a 1 + a 101 ) = 2 101 ( 1 + 201 ) = 2 101 ( 202 ) = 101 × 101 = 10201 .
Comparing the Sums Finally, we compare the sum of the first 100 even natural numbers ( S 100 = 10100 ) with the sum of the first 101 odd natural numbers ( S 101 = 10201 ). Since 10100 < 10201 , the sum of the first 100 even natural numbers is less than the sum of the first 101 odd natural numbers.
Examples
Arithmetic sequences are useful in many real-life situations. For example, if you save $100 each month, the total amount you save over time forms an arithmetic sequence. Understanding how to calculate the sum of an arithmetic sequence can help you predict your total savings after a certain number of months. Similarly, if a theater has rows with an increasing number of seats, arithmetic sequences can help determine the total seating capacity.
The 100th even natural number is 200, and the sum of the first 100 even natural numbers is 10100. This sum is less than the sum of the first 101 odd natural numbers, which is 10201. Therefore, the answer is 'less than'.
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