$7\mid 5^n +1$

We note that $7 \mid 5^n + 1 \implies 5^n \equiv -1 \text{ (mod 7)}$ . Consider $5^n \equiv (7-2)^n \equiv (-2)^n \text{ (mod 7)}$ . For the first few $n$ , we have $(-2)^n = -2, 4, -1, 2, -4, 1, -2, ....$ We note that $(-2)^n$ has a period of 6 and $(-2)^n = -1$ , when $n \bmod 6 = 3$ . For $n \le 1000$ , the acceptable $n=3,9,15,... 999$ , a total of $\boxed{167}$ of them.

Computer science solution cuz number theory ain't my cup of tea xD

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n=1
s=0
while n <= 1000:
    if (5**n + 1) % 7 == 0:
        s += 1
    n += 1
print(s)