The question is below

See below.

Step-by-step explanation:

a.

The first figure has 1 square. The second figure has a column of 2 squares added to the left. The third figure has a column of 3 squares added to the left. Each new figure has a column of squares added to the left containing the same number of squares as the number of the figure.

b.

Figure 10 has 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 squares.

c.

The formula for adding n positive integers starting at 1 is:

1 + 2 + 3 + ... + n = n(n + 1)/2

For figure 55, n = 55.

n(n + 1)/2 = 55(56)/2 = 1540

d.

Let's use the formula set equal to 190 and solve for n. If n is an integer, then we can.

n(n + 1)/2 = 190

n(n + 1) = 380

We know that 380 = 19 * 20, so n = 19.

yes

e.

Use the formula above,

S = n(n + 1)/2, where S is the sum.

f.

n(n + 1) = 1478

38 * 39 = 1482

37 * 38 = 1406

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step-by-step explanation:

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