In: Advanced Math
For the following exercises, write the first five terms of the arithmetic series given two terms.
a13 = −60, a33 = −160
Consider an arithmetic sequence with a13 = -60 and a33 = -160
Use the formula for nth term of an arithmetic sequence,
an = a1 + (n – 1)d ...... (1)
Substitute n = 13 and a13 = -60 in formula (1),
-60 = a1 + (13 – 1)d
-60 = a1 + 12d ...... (2)
Substitute n = 33 and a33 = -160 in formula (1),
-160 = a1 + (33 – 1)d
-160 = a1 + 32d ...... (3)
Subtract equation (2) -60 = a1 + 12d from (3) -160 = a1 + 32d,
-160 = a1 + 32d
-60 = a1 + 12d
-100 = 20d
Division by 20 gives,
d = -100/20
d = -5
Substitute d = -5 in equation (2) -60 = a1 + 12d,
-60 = a1 + 12 × (-5)
a1 = -60 + 60
a1 = 0
Compute second term of the sequence as follows:
a2 = a1 + d
= 0 + (-5)
= -5
Compute third term of the sequence as follows,
a3 = a2 + d
= -5 +(-5)
= -10
Compute forth term of the sequence as follows:
a4 = a3 + d
= -10 + (-5)
= -15
Compute fifth term of the sequence as follows:
a5 = a4 + d
= -15 + (-5)
= -20
Therefore, first five terms of given arithmetic sequence are {0, -5, -10, -15, -20}.