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Теория: 03 Характеристическое свойство арифметической прогрессии

Задание

Дана арифметическая прогрессия: \(\displaystyle 2, 4, 6, 8, …\)

Сравните \(\displaystyle (a_{10} + a_{14})\) и \(\displaystyle (a_{11} + a_{13}){\small .}\)

\(\displaystyle (a_{10} + a_{14})\) \(\displaystyle (a_{11} + a_{13}){\small .}\)

Решение

По условию

\(\displaystyle a_1 = 2{ \small ,}\,a_2 = 4{ \small .}\)

Следовательно, можно найти разность \(\displaystyle d\) данной прогрессии:

\(\displaystyle d = 4 - 2{ \small ,}\)

\(\displaystyle d = 2{ \small .}\)

Тогда 

\(\displaystyle a_{10} = a_1 + 9d{ \small ,}\)

\(\displaystyle a_{10} = 2 + 9 \cdot 2{ \small ,}\)

\(\displaystyle a_{10} = 20{ \small .}\)

Точно так же найдем \(\displaystyle a_{11}{ \small ,}\,a_{13}\) и \(\displaystyle a_{14}{\small : }\)

\(\displaystyle a_{11} = a_1 + 10d{ \small ,}\)

\(\displaystyle a_{11} = 2+10\cdot 2\)

\(\displaystyle a_{11} = 22{ \small .}\)

\(\displaystyle a_{13} = a_1 + 12d{ \small ,}\)

\(\displaystyle a_{13} = 2+ 12\cdot 2{ \small ,}\)

\(\displaystyle a_{13} = 26{ \small .}\)

\(\displaystyle a_{14} = a_1 + 13d{ \small ,}\)

\(\displaystyle a_{14} = 2+ 13\cdot 2{ \small ,}\)

\(\displaystyle a_{14} = 28{ \small .}\)

Теперь выполним сравнение:

\(\displaystyle a_{10} + a_{14}= 20 + 28{ \small ,}\)

\(\displaystyle a_{10} + a_{12}= 48{ \small ;}\)

\(\displaystyle a_{11} + a_{13}= 22 + 26{ \small ,}\)

\(\displaystyle a_{11} + a_{13} = 48{ \small .}\)

Значит, искомые величины равны.

Ответ: \(\displaystyle (a_{10} + a_{14})\boldsymbol{=}(a_{11} + a_{13}){\small .}\)