#### A sequence a_1, a_2, a_3 ... , is defined by a_1 = k, a_{n+1} = 3a_n + 5, n ≥ 1, where k is a positive integer.

i) Find \sum_{r=1}^4a_r in terms of k.

(ii) Show that \sum_{r=1}^4a_r is divisible by 10.

Anonymous

0

i) Find \sum_{r=1}^4a_r in terms of k.

(ii) Show that \sum_{r=1}^4a_r is divisible by 10.

Krishna

0

(i)

Step 1: Examine the summation

EXAMPLE: \sum_{r=1}^{5} a_r = a_1 + a_2 + a_3 + a_4 + a_5

Step 2: Make sure that the given set of numbers arranged in some particular order. Because the question says that the set of numbers in sequence.

Step 3: Explore the given rule

EXAMPLE: x_{n+1} = ax_n - 3

Succeeding term = a (preceding term) - 3

Step 4: According to the given rule substitute the (n)values.

[NOTE: To find the twenty-first term, replace n(any variable represents position) by 21. based up on the rule it(n) may change to lower value or higher value]

EXAMPLE: For an attempt to find the x_2 substitute n=1 in the given rule

x_{n+1} = ax_n - 3

x_2 = ax_1 - 3

Substitute the values x_1 [ take x_1 = 1 since n >1), Some times it mention

in the question]

x_ 2 = a(1) - 3

x_2 = a - 3

Step 5: Simplify further

(Apply the BODMAS rules)

Step 6: Repeat the same steps to calculate all the values of a_1, a_2, a_3, a_4, a_5 ...and add them

(ii)

Step 7: Divide the summation value by 10