How many terms of the AP 24, 21, 18, … must be taken so that their sum is 78?

Class 10Arithmetic ProgressionsAlgebra — Sequences

Step-by-Step Solution

a = 24, d = - 3. \frac{n}{2} [48 + (n - 1) (- 3)] = 78

→

n (51 - 3 n) = 156

→

3 n^{2} - 51 n + 156 = 0

→

n^{2} - 17 n + 52 = 0

→

(n - 4) (n - 13) = 0

→

n = 4

or 13. Both are valid.

Answer: A) 4 or 13

Between two numbers a and

b

, you can insert

k

'arithmetic means' so that the resulting sequence is in AP.

**Setup**: a,

m

₁,

m

₂, …,

m_{k}, b

. Total of

k + 2

terms in AP, so:
- First term =

a

, last term =

b

.
- Common difference

d = (b - a) / (k + 1)

(because there are

k + 1

'gaps').

**Example**: insert 3 arithmetic means between 4 and 16.

d = (16 - 4) / (3 + 1) = 3

. Means: 7, 10, 13. Check: 4, 7, 10, 13, 16 —

d = 3

✓.

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