Question

Consider the arithmetic sequence 100, 93, 86,... . Common difference = -7. What is the remainder obtained when each positive term of this sequence is divided by 7?

Answer 1

Hi,

2

Explanation:

Let's assume u the sequence, r the remainder modulo 7.

`u_1=100, \ 100 \equiv 2 [7] \\u_2=u_1-1*7=93,\ 93 \equiv 2 [7] \\ u_3=u_1-2*7=86,\ 86 \equiv 2 [7] \\...\\u_(14)=u_1-13*7=100-91=9,\ 9 \equiv 2 [7] \\u_(15)=u_1-14*7=100-98=2,\ 2 \equiv 2 [7] \\u_(16)\ is\ negative\\\\`

Answer 2

Therefore, the remainder obtained when each positive term of this sequence is divided by 7 is 2.

Explanation:

To find the remainder obtained when each positive term of the arithmetic sequence is divided by 7, we need to observe the pattern in the sequence.

The first term of the sequence is 100, and the common difference is -7. This means that to find the next term, we subtract 7 from the previous term.

100 - 7 = 93

93 - 7 = 86

We can continue this pattern to find the subsequent terms of the sequence.

To find the remainder when each positive term is divided by 7, we can divide each term by 7 and observe the remainder.

100 divided by 7 is 14 with a remainder of 2.

93 divided by 7 is 13 with a remainder of 4.

86 divided by 7 is 12 with a remainder of 2.

We can see that there is a repeating pattern in the remainders. The remainder sequence is 2, 4, 2, 4, ...