O lvl A Maths: Indices and Logarithms

Show that 5ⁿ + 5ⁿ⁺¹ + 5ⁿ⁺² is divisible by 31 for all positive integer values of n

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Answer:

5ⁿ + 5ⁿ⁺¹ + 5ⁿ⁺²
= 5ⁿ + 5ⁿ 5¹ + 5ⁿ 5²
= 5ⁿ ( 1 + 5¹ + 5² )
= 5ⁿ (31)

Hence, 5ⁿ + 5ⁿ⁺¹ + 5ⁿ⁺² is divisible by 31 for all positive integer values of n (shown).