Read, analyze and solve. 1. How many numbers between 7 and 501 are divisible by 5? 2. How many numbers are divisible by 3 between 100 and 1000? 3. In the sequence 2, 6, 10, ...; find n if the nth term is 102. 4. In the sequence: 10, 7, 4, ...; what term has a value of -296?
Answer (1)
Let's solve each part of the question step-by-step:
How many numbers between 7 and 501 are divisible by 5?
To find the numbers divisible by 5, let's first identify the smallest and largest numbers divisible by 5 within the range:
The smallest number greater than or equal to 7 that is divisible by 5 is 10.
The largest number less than or equal to 501 that is divisible by 5 is 500.
These numbers form an arithmetic sequence where the first term a 1 = 10 and the common difference d = 5 .
To find the total number of terms n , use the formula for the nth term of an arithmetic sequence, a n = a 1 + ( n − 1 ) ⋅ d .
Set a n = 500 :
500 = 10 + ( n − 1 ) ⋅ 5 490 = ( n − 1 ) ⋅ 5 n − 1 = 5 490 = 98 n = 99
Therefore, there are 99 numbers between 7 and 501 that are divisible by 5.
How many numbers are divisible by 3 between 100 and 1000?
Let's use the same approach:
The smallest number greater than or equal to 100 that is divisible by 3 is 102.
The largest number less than or equal to 1000 that is divisible by 3 is 999.
These numbers also form an arithmetic sequence with first term a 1 = 102 and common difference d = 3 .
Solve for n using the nth term formula:
999 = 102 + ( n − 1 ) ⋅ 3 897 = ( n − 1 ) ⋅ 3 n − 1 = 3 897 = 299 n = 300
There are 300 numbers between 100 and 1000 that are divisible by 3.
In the sequence 2, 6, 10, ...; find n if the nth term is 102.
This sequence is arithmetic with first term a 1 = 2 and common difference d = 4 .
Use the nth term formula:
a n = a 1 + ( n − 1 ) ⋅ d Set a n = 102 :
102 = 2 + ( n − 1 ) ⋅ 4 100 = ( n − 1 ) ⋅ 4 n − 1 = 25 n = 26
Thus, the 26th term is 102.
In the sequence 10, 7, 4, ...; what term has a value of -296?
This sequence is arithmetic with first term a 1 = 10 and common difference d = − 3 .
Set a n = − 296 :
− 296 = 10 + ( n − 1 ) ⋅ ( − 3 ) − 296 = 10 − 3 ( n − 1 ) − 306 = − 3 ( n − 1 ) 306 = 3 ( n − 1 ) 102 = n − 1 n = 103
Therefore, the 103rd term of the sequence is -296.
