The statement in the question appears truncated, but the natural and standard interpretation is:

$A=\{1,a_1,a_2,\dots,a_{18},77\}$ is a set of integers in A.P.

We are asked to find $1+a_1+a_2+\cdots+a_{18}.$

Since there are 20 terms in the set, $1, a_1, a_2, \dots, a_{18}, 77$ form an arithmetic progression with first term $1$ and last term $77$.

1. Number of terms

There are:

• first term: $1$
• middle terms: $a_1,a_2,\dots,a_{18}$ (18 terms)
• last term: $77$

So total number of terms is $n=20.$

2. Common difference

For an arithmetic progression, $l=a+(n-1)d.$

Here, $77=1+19d$ so $19d=76$ $d=4.$

Thus the full A.P. is $1,5,9,13,\dots,73,77.$ Hence $a_1=5,\ a_2=9,\dots,\ a_{18}=73.$

3. Required sum

We need $1+a_1+a_2+\cdots+a_{18},$ which is the sum of the first 19 terms of the A.P.: $1,5,9,\dots,73.$

Using sum formula, $S_{19}=\frac{19}{2}(\text{first}+\text{19th term})=\frac{19}{2}(1+73).$

So, $S_{19}=\frac{19}{2}\cdot 74=19\cdot 37=703.$

4. Comparison with stored answer

My derived answer is $\boxed{703}.$

The stored correct answer is $702$, which does not match.

So I disagree with the stored answer. The likely reason is either:

1. a typo in the question statement, or
2. a typo in the stored answer.

Under the standard interpretation of the given set as an arithmetic progression, the correct value is $703$.