Question

All the term of an A. P. are natural numbers and the sum of the first 20 terms
is greater than 1072 and less than 1162. If the sixth term is 32 then common difference and first term is​

Answer 1

Answer:

Required value of first term is 7 and common difference between the terms is 5.

Step-by-step explanation:

It is given that the terms of A.P. are natural numbers.

It means : 1st term, 2nd term , 3rd term, and all the terms are natural terms.

Let the first term of this sequence be a and the common difference between the terms be d.

First term belongs of natural numbers, that implies a belongs to natural numbers and can't be any fraction or decimal number.

Given,

6th term of the AP is 32

= > a + ( 6 - 1 )d = 32

= > a + 5d = 32

= > 5d = 32 - a

= > d = ( 32 - a ) / 5 ... ( 1 )

Sum of first 20 numbers is greater than 1072 and less than 1162.

= > 1072 < S$_{20}$ < 1162

From the properties of arithmetic sequences :

• Sum of n terms = ( n / 2 ) [ 2a + ( n - 1 )d ], where n is the number of terms, a is the first term and d is the common difference between the terms.

Thus, { continued }

= > 1072 < ( 20 / 2 ){ 2a + ( 20 - 1 )d } < 1162

= > 1072 < 10( 2a + 19d ) < 1162

= > 1072 < 10[ 2a + 19( 32 - a ) / 5 ] < 1162 { from ( 1 ) }

= > 1072 < ( 10 / 5 ) [ 10a + 19( 32 - a ) ] < 1162

= > 1072 < 2( 10a + 608 - 19a ) < 1162

= > 536 < 608 - 9a < 581

= > 536 < 608 - 9a And 608 - 9a < 581

= > 9a < 608 - 536 And 608 - 581 < 9a

= > 9a < 72 And 27 < 9a

= > a < 72 / 9 And 27 / 9 < a

= > a < 8 And 3 < a

= > 3 < a < 8

Hence value of a lies between 3 and 8 , it can be 4 or 5 or 6 or 7 , since a is a natural number.

• When a is 4, 6th term = 32 = > 4 + 5d = 32 = > d = 28 / 5 .
• When a is 5, 6th term = 32 = > 5 + 5d = 32 = > d = 27 / 5
• When a is 6, 6th term = 32 = > 5 + 5d = 32 = > d = 26 / 6.

In the above cases, d is not a natural number, so the 7th term or 5th won't be a natural number, but all the terms are natural numbers. So, a can't be equal to 4 or 5 or 6.

• When a is 7, 6th term = 32 = > 7 + 5d = 32 = > d = 5

In this case, d is also a natural number, so we can say that all the term will be natural.

Hence the required value of first term is 7 and common difference between the terms is 5.