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More correctly, it's what you get after you do addition. Consider "3+5=8". The action of calculating "3+5" is addition. "8" is the sum. At least, in the context that die Summe means. Some English-speakers would also call arithmetical calculations themselves "sums" (e.g. "3+5"), however in German that would be (eine) Rechenaufgabe (lit. "calculation task") according to Pons.
"Sum" can also be a verb in English (synonymous with "add [up]"), but not in German. Other words would be used for that.
I learned about seasons in another language app, and this sounds so close to "sommer".
I could not hear her saying "die Summe" sounded more like "die Söhne" (this was an "exercise," so many random phrases). Those are very close in sound, and with no context... Gah!
And I see that I am not the only one who had a hard time hearing it correctly.
[later] Tip: It's in the vowel sound. Listen for "oo" vs "oh"