Consider the arithmetic sequence 5,7,9,11, …. Derive a formula for (i) the nth term and (ii) the sum to n terms. (iii) Hence find the sum of the first 20 terms.
We can easily identify the first term (5)
The common difference can be found by subtracting the nth term from the (n+1)th term
7-5=9-7=11-9=2
Therefore:
U1=5 and d=2
The IB formula booklet provides the general formula for the nth term and the sum to n terms. Substitute the previously found values into these formulae.
Un=U1 + (n-1)d
Sn=n/2(2u1+(n-1)d)
(i)
Un=U1 + (n-1)d
Substitute in values of u1 and d
Un=5+(n-1)2
Simplify the result by expanding brackets
Un=5+2n-2
Un=2n+3
(ii)
Sn=n/2(2u1+(n-1)d)
Substitute in values of u1 and d
Sn=n/2(2(5)+(n-1)2)
Sn=n/2(10+2n-2)
Sn=5n+n2-n
Sn=4n+n2
(iii)
Substitute n=20 into the formula from (ii)
Sn=4n+n2
Sn=4(20)+(20)2
Solve
Sn=80+400=480
JN
