The description and use of two arithmetick instruments together with a short treatise, explaining and demonstrating the ordinary operations of arithmetick, as likewise a perpetual almanack and several useful tables : presented to His most excellent Majesty Charles II ... / by S. Morland.

IV. Reflection.

IN all Continued Arithmetical Progressions, (how many soever the Terms be) the Sum of the Ex∣treams is equal to the Sum of any two of the other Terms equidistant from the Extreams, and to the double of the Middle Term, in case the Number of Terms be odd.

Let the Number of Terms be Seven.

viz. 3, 6, 9, 12, 15, 18, 21, and the Common Difference (3).

First by the Hypothesis.

The four Numbers (3, 6, 18, 21) are four Pro∣portionals.

Therefore by the fourth Reflection.

Their two sums, viz

• 3 more by 21
• 6 more by 18

are Equal.

Secondly, by the same Hypothesis.

The four Numbers (6, 9, 15, 18,) are four Proportionals.

Therefore by the fourth Reflection.

Their two sums, viz.

• 6 more by 18
• 9 more by 15

are Equal.

Lastly, by the same Hypothesis.

The three Numbers (9, 12, 15) are three Proportionals.