Why output is different though expressions are same in lambda expression?

I have problem about lambda expression like this:

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std; 

int sum(vector<int>& v){
    int total = 0;
    auto lambda = for_each(v.begin(), v.end(), [&total](int n){total += n;});
    lambda; // lambda expression doesn't work.
    for_each(v.begin(), v.end(), [&total](int n){total += n;}); // work same as I intended.
    return total;
}

int main(void){
    vector<int> v = {1, 2, 3, 4, 5};
    cout << sum(v) << endl; // 30 (I think this should be 45.)
}

I thought that lambda; could do same things like for_each algorithm. Why lambda; doesn’t works?

>Solution :

according to cppreference, the for_each(...) call returns the UnaryFunction which was passed to the function.

In this case, the for_each returns the UnaryFunction

[&total](int n){total += n;}

a lambda(5) would increase the total value by 5.

A solution would be to put the for_each call in a separate function – which is actually a "sum" function.
This is already done by the std:accumulate function

std::accumulate(v.begin(), v.end(), 0);