Consider the arithmetic sequence 100, 109, 118 · · ·
a) What is the common difference ?
b) What is the remainder on dividing the terms of the arithmetic sequence by 9
c) Which is the first four digit term of this sequence ?
d) Find the sum of all three digit terms of this sequence
Answers
SOLUTION
GIVEN
Consider the arithmetic sequence 100, 109, 118 · · ·
TO DETERMINE
a) The common difference
b) The remainder on dividing the terms of the arithmetic sequence by 9
c) The first four digit term of this sequence
d) The sum of all three digit terms of this sequence
EVALUATION
Here the given arithmetic sequence is
100, 109, 118 · · ·
First term = a = 100
Common Difference = d = 109 - 100 = 9
a) The common difference = 9
b) The nth term of the sequence
= a + (n - 1)d
= 100 + 9( n - 1 )
= 100 + 9n - 9
= 9n + 91
= 9( n + 10 ) + 1
Now 9( n + 10 ) is divisible by 9
Hence the required Remainder = 1
c) The smallest 4 digit number = 1000
By the given condition
9n + 91 ≥ 1000
⇒ 9n ≥ 909
⇒ n ≥ 101
The first four digit term of this sequence
= 101th term of the sequence
= 1000
d) The sum of all three digit terms of this sequence
= 100 + 109 + 118 + . . . + 100th term
= 100 + 109 + 118 + . . . + 991
= 54550
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Step-by-step explanation:
SOLUTION
GIVEN
Consider the arithmetic sequence 100, 109, 118 · · ·
TO DETERMINE
a) The common difference
b) The remainder on dividing the terms of the arithmetic sequence by 9
c) The first four digit term of this sequence
d) The sum of all three digit terms of this sequence
EVALUATION
Here the given arithmetic sequence is
100, 109, 118 · · ·
First term = a = 100
Common Difference = d = 109 - 100 = 9
a) The common difference = 9
b) The nth term of the sequence
= a + (n - 1)d
= 100 + 9( n - 1 )
= 100 + 9n - 9
= 9n + 91
= 9( n + 10 ) + 1
Now 9( n + 10 ) is divisible by 9
Hence the required Remainder = 1
c) The smallest 4 digit number = 1000
By the given condition
9n + 91 ≥ 1000
⇒ 9n ≥ 909
⇒ n ≥ 101
The first four digit term of this sequence
= 101th term of the sequence
= 1000
d) The sum of all three digit terms of this sequence
= 100 + 109 + 118 + . . . + 100th term
= 100 + 109 + 118 + . . . + 991
= 54550
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