Multiple choice

The number $a_{n}=6^{n}-5n$ for $n=1,2,3,....$ when divided by $25$ leave the remainder

  1. $9$
  2. $7$
  3. $3$
  4. $1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

a_n = 6^n - 5n. For n=1, a_1 = 6-5=1. 1 mod 25 = 1. For n=2, a_2 = 36-10=26. 26 mod 25 = 1. For n=3, a_3 = 216-15=201. 201 mod 25 = 1. The remainder is always 1.

AI explanation

Test the given expression for a general value of n to find the pattern of remainders. For n equals 1, the expression is 6 to the power of 1 minus 5 times 1, which is 1. For n equals 2, the expression is 6 to the power of 2 minus 5 times 2, which is 26; when 26 is divided by 25, the remainder is 1. Because the remainder is always 1 for any positive integer n, the correct answer is 1.

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