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(b) By inspection and induction,

Tn = ( n + 4 ) ² –  (4( n + 3) + 2)

= n² + 8n + 16 – ( 4n + 14)

= n² + 4n + 2      proved

(c) 962 = n² + 4n + 2 

n² + 4n- 960 = 0                      

=   (-4 ±√ ( 16 + 4(960)))/2

= -2 ± (√ (3856))/2

Since √ (3856) not integer,  962 is not a term of the sequence

(d) Tn – Tn-1 = n² + 4n + 2  – (n-1)² – 4(n-1) -2

= n² + 4n + 2 – n² + 2n  -1 – 4n + 4 – 2

= 2n + 3

(e) As difference between terms is 2n+3 from (d), since 2n must be even , then 2n+ 3 must be odd.

 

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