Answer:
16
Step-by-step explanation:
Given the expression 1+2+3 +...+x=120, this means that the sum of the arithmetic sequence 1+2+3 +...+x is 120.
First we need to find the xth term of the arithmetic sequence 1+2+3 +...+x
Sum of an arithmetic seq Sx = x/2(2a+(x-1)d)
a = first term
d = common difference
x is the number of terms
from the sequence, a = 1 and d = 1
Sx = x/2(2(1)+(x-1)*1)
Sx = x/2(2+x-1)
Sx = x/2(1+x)
Sx = x²+x/2
Substitutig thid into the question to calculate x;
x²+x/2 = 120
x²+x = 240
x²+x-240 = 0
x = 1±√1²-4(-240)/2
x = 1±√961/2
x = 1±31/2
x = 1+31/2 or 1-31/2
x = 32/2 or -30/2
x = 16 or -15
Since we are to look for the positive integer of x, then x is 16
