In one sentence, describe what the following article is about: In the formula, xn{\displaystyle x_{n}} = the term in the sequence you are trying to find, n{\displaystyle n} = the position number of the term in the sequence, and {\displaystyle \phi } = the golden ratio. This is a closed formula, so you will be able to calculate a specific term in the sequence without calculating all the previous ones. This formula is a simplified formula derived from Binet's Fibonacci number formula. The formula utilizes the golden ratio ({\displaystyle \phi }), because the ratio of any two successive numbers in the Fibonacci sequence are very similar to the golden ratio. The n{\displaystyle n} represents whatever term you are looking for in the sequence. For example, if you are looking for the fifth number in the sequence, plug in 5. Your formula will now look like this: x5{\displaystyle x_{5}}=5(1)55{\displaystyle {
rac {\phi ^{5}-(1-\phi )^{5}}{\sqrt {5}}}}. You can use 1.618034 as an approximation of the golden ratio. For example, if you are looking for the fifth number in the sequence, the formula will now look like this: x5{\displaystyle x_{5}}=(1.618034)5(11.618034)55{\displaystyle {
rac {(1.618034)^{5}-(1-1.618034)^{5}}{\sqrt {5}}}}. Remember to use the order of operations by completing the calculation in parentheses first: 11.618034=0.618034{\displaystyle 1-1.618034=-0.618034}. In the example, the equation becomes x5{\displaystyle x_{5}}=(1.618034)5(0.618034)55{\displaystyle {
rac {(1.618034)^{5}-(-0.618034)^{5}}{\sqrt {5}}}}. Multiply the two parenthetical numbers in the numerator by the appropriate exponent. In the example, 1.6180345=11.090170{\displaystyle 1.618034^{5}=11.090170}; 0.6180345=0.090169{\displaystyle -0.618034^{5}=-0.090169}. So the equation becomes x5=11.090170(0.090169)5{\displaystyle x_{5}={
rac {11.090170-(-0.090169)}{\sqrt {5}}}}. Before you divide, you need to subtract the two numbers in the numerator. In the example, 11.090170(0.090169)=11.180339{\displaystyle 11.090170-(-0.090169)=11.180339}, so the equation becomes x5{\displaystyle x_{5}}=11.1803395{\displaystyle {
rac {11.180339}{\sqrt {5}}}}. The square root of 5, rounded, is 2.236067. In the example problem, 11.1803392.236067=5.000002{\displaystyle {
rac {11.180339}{2.236067}}=5.000002}. Your answer will be a decimal, but it will be very close to a whole number. This whole number represents the number in the Fibonacci sequence. If you used the complete golden ratio and did no rounding, you would get a whole number. It's more practical to round, however, which will result in a decimal. In the example, after using a calculator to complete all the calculations, your answer will be approximately 5.000002. Rounding to the nearest whole number, your answer, representing the fifth number in the Fibonacci sequence, is 5.
Summary: Set up the formula xn{\displaystyle x_{n}}=n(1)n5{\displaystyle {
rac {\phi ^{n}-(1-\phi )^{n}}{\sqrt {5}}}}. Plug the number for n{\displaystyle n} into the formula. Substitute the golden ratio into the formula. Complete the calculations in parentheses. Calculate the exponents. Complete the subtraction. Divide by the square root of 5. Round to the nearest whole number.

In one sentence, describe what the following article is about: You can't just have a pond and attract dragonflies. You need water plants in the pond to help attract them. Dragonflies love tall plants! The dragonflies will lay their eggs in the plants, and the larvae will live in them until they are grown. Then, they will use the tall plants to perch on. Stock the pond with both submerged and floating plants. They will use the underwater plants when they are larvae, and perch on the tall plants when they are adults. Try getting eelgrass, fanwort, hornwort, anacharis, wild celery, corkscrew rush, blue flag iris, parrot's feather, pondweed, water lilies, and lotus flowers. You can find water plants at garden centers and online. Place shrubs around the edge of the pond to give the dragonflies more places to land. This also makes your pond even more beautiful and appealing. The dragonflies will have more places to perch and live. You can plant border plants and shrubs. For example, you can try lobelia, seedbox, or button bush. You can also let the natural grass and brush around the pond grow to give the dragonflies more vegetation. Rocks in and around your pond will make it even more beautiful. Dragonflies also love to perch on warm rocks, especially flat rocks. Place rocks in your pond and around the edges to give the dragonflies plenty of places to land. You can try a mixture of light and dark rocks. The dragonflies may be attracted to one over the other. Dragonflies like sunshine, so they will be more attracted to a pond that is out in the open with full sunshine in the middle of the day than one covered by the shade of tree branches. While you are waiting for your plants to grow tall enough for dragonflies to perch on, place sticks in your pond. This gives the dragonflies somewhere to land. You can try sticks that you get from nearby trees or bamboo stakes for plants and vegetables.
Summary:
Add water plants to your pond. Surround the pond with shrubs. Spread rocks around your pond. Make sure the pond gets plenty of sun. Place sticks in the middle of your pond.