This paper develops a finite framework for studying sums of prime numbers through consecutive prime gaps. The motivation comes from Ramanujan’s influence on arithmetic decompositions, partition methods, and summation ideas, while the argument remains within ordinary finite number theory. We study the partial sum of the first n primes and prove an exact identity that writes this sum as a baseline term from the initial prime together with a weighted accumulation of consecutive prime gaps. Each gap receives a weight equal to the number of later primes affected by that gap. The same identity is then expressed geometrically by encoding each weighted gap as the slope of a right triangle. Exact numerical examples verify the formula, and the asymptotic discussion shows that the decomposition has the same leading scale as the classical growth predicted by the prime number theorem. This paper also separates finite identities from regularized infinite summations. This distinction is essential: the finite formula is exact, while divergent infinite prime sums require a separate summation theory and cannot be treated as ordinary convergent series.