The nth term of a geometric series is tn and the common ratio is r. where r>0 Given that t1 = 1. (a) Write down an expression in terms of r and n for tn.Given also that tn+t(n+1)=t(n+2). (b) Show that r=1+√5/2. (c) Find the exact value of t4 giving your answer in the form of f+g√h, where f, g and h are integers.
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Given a geometric sequence with the first term a1 and the common ratio r , the nth (or general) term is given by
an=a1⋅rn−1 .
Example 1:
Find the 6th term in the geometric sequence 3,12,48,... .
a1=3, r=123=4a6=3⋅46−1=3⋅45=3072
Example 2:
Find the 7th term for the geometric sequence in which a2=24 and a5=3 .
Substitute 24 for a2 and 3 for a5 in the formula
an=a1⋅rn−1 .
a2=a1⋅r2−1→24=a1ra5=a1⋅r5−1→ 3=a1r4
Solve the firstequation for a1 : a1=24r
Substitute this expression for a1 in the second equation and solve for r .
3=24r⋅r43=24r318=r3 so r=12
Substitute for r in the first equation and solve for a1 .
24=a1(12)48=a1
Now use the formula to find a7 .
a7=48(12)7−1=48⋅164=34