maths 高手請答~

Let the nth term of the arithmetic sequence be
T(n) = a1 + (n - 1)d
where a1 = 1st term, d = common difference

T(4) + T(8) + T(12) + T(16) =224
[a1 + (4 - 1)d] + [a1 + (8 - 1)d] + [a1 + (12 - 1)d] + [a1 + (16 - 1)d] = 224
(a1 + 3d) + (a1 + 7d) + (a1 + 11d) + (a1 + 15d) = 224
4a1 + 36d = 224
a1 + 9d = 56 ................(1)

The value of an arithmetic series (sum) consiting of n terms a1, a2, ... , anis given by the formula:
Sn = a1 + a2 + ... + an = n[2a1 + (n -1)d] /2

sum of the first 19 terms of this sequence
= S19
= a1 + a2 + ... + a19
= (19)[2a1 + (19 -1)d] /2
= (19)[2a1 + 18d] /2
= (19)(a1 + 9d) ...............................But from (1), a1 + 9d = 56
= (19)(56)
= 1064