Why is 3/4 [ 1 + 3 ⁿ⁻¹ (2n-3+4n - 4n) ] -3 = 3/4 [ 1 + 3 ⁿ ( 2n- 1) ] -3 - n 3 ⁿ

Starting from the left hand side.

Why is LHS = RHS ?

Thanks

3/4 [ 1 + 3ⁿ⁻¹ (2n − 3 + 4n − 4n) ] − 3

= 3/4 [ 1 + 3ⁿ⁻¹ (2n − 3) ] − 3

= 3/4 [ 1 + 3ⁿ/3 (2n − 3) ] − 3

= 3/4 [ 1 + 3ⁿ (2/3 n − 1) ] − 3

= 3/4 [ 1 + 3ⁿ (2/3 n − 1) + 3ⁿ (4/3 n) ] − 3 − 3/4 [3ⁿ (4/3 n)]

= 3/4 [ 1 + 3ⁿ (2/3 n − 1 + 4/3 n) ] − 3 − 3/4 [ 4/3 n 3ⁿ ]

= 3/4 [ 1 + 3ⁿ (6/3 n − 1) ] − 3 − n 3ⁿ

= 3/4 [ 1 + 3ⁿ (2n − 1) ] − 3 − n 3ⁿ

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